Mastering Circle Tangent Line Theorems and Problems
in Geometry
Tangent lines embrace,
Circle's edge with gentle touch,
Theorems come alive.
Problems find solutions sweet,
Tangents guide the way to truth.
In geometry, a tangent line is a
line that intersects a circle at exactly one point,
known as the point of tangency. Here are some important concepts related
to tangent lines and circles:
-
Tangent: A line that
intersects a circle at exactly one point.
-
Point of
tangency: The point where a tangent line intersects a
circle.
-
Radius: A line segment that connects the
center of a circle to a point on the circle.
-
Tangent
theorem: If a line is tangent to a circle, then the line
is perpendicular to the radius at the point of tangency.
-
Secant: A line that intersects a circle at two points.
-
Chord: A line segment that connects two points on a
circle.
-
Equal tangent theorem: the tangent segments drawn to a circle from an external point are equal in length.
-
Exterior angle theorem: The measure of an angle formed by a tangent
line and a secant line is equal to the difference between the measures
of the intercepted arcs.
-
Secant-tangent theorem: If a tangent and a
secant are drawn from the same external point, then the product of the
lengths of the secant and its external segment is equal to the square
of the length of the tangent.
-
Angle between tangent and chord
theorem: The measure of the angle formed by a tangent line and a chord
is half the measure of the intercepted arc on the opposite side of the
chord.
Understanding these concepts is essential in
solving geometry problems involving circles and tangent
lines.
|
![Tangent Line to a Circle](tangent_line_circle_i.jpg) |
Circle Tangent Line - Index
|
![Incenter, Incircle, Index](../center/incenter_incircle_17.jpg) |
Incircle of a triangle.
Index.
Tangent line to a circle
|
![Excenter, Excircle, Index](../center/excenter_excircle_17.jpg) |
Excircle.
Index.
Tangent line to circle.
|
![Lemoine Line Theorem](../geometry/lemoine-line-theorem-circle-html5-animation-for-ipad-nexus-1.jpg) |
Lemoine Line Theorem,
Interactive illustration.
HTML5 Animation for iPad and Nexus |
![Dynamic Geometry: Lemoine Line of a Triangle. HTML5 Animation for Tablets (iPad, Nexus..)](../geogebra/lemoine-line-triangle-html5-animation-ipad-nexus-1.jpg) |
GeoGebra, Dynamic Geometry:
Lemoine Line
of a Triangle.
HTML5 Animation for Tablets (iPad, Nexus..) |
![Geometry Problem 1568](../school-college/6/p1568-proving-concyclicity-in-triangle-with-incircle-and-tangent-circle-problem-solving-tutoring-1.jpg) |
Geometry Problem 1568:
Concyclicity of Points B, D , H, J in a Triangle ABC with an Incircle and a Tangent Circle. |
![Geometry Problem 1567](../school-college/6/p1567-bases-isosceles-trapezoid-circumscribed-circle-chord-distance-measurement-tangent-tutoring-1.jpg) |
Geometry Problem 1567:
Finding Tangent Distances in a Circumscribed Isosceles Trapezoid. |
![Geometry Problem 1556](../school-college/6/p1556-right-triangle-inscribed-circle-properties-and-length-of-FG-problem-solving-1.jpg) |
Geometry Problem 1556: Right Triangle ABC and Inscribed Circle. The problem involves circle, chords, tangent, perpendicular lines, and congruence |
![Geometry Problem 1555](../school-college/6/p1555-circle-chord-intersect-geometry-problem-solving-1.jpg) |
Geometry Problem 1555: Find Length of DE. The problem involves chords, tangent, circles, and intersections of line |
![Geometry Problem 1549](../school-college/6/p1549-triangle-incircle-tangent-inscribed-circle-angle-calculation-1.jpg) |
Geometry Problem 1549: Unraveling the Geometric Mystery: Calculating Angle
BGE with the Incircle and Tangent in Triangle ABC |
![Geometry Problem 1547](../school-college/6/p1547_circle_radius_4_tangents_AB_AC_points_of_tangency_distance_calculation_from_B_to_AC-1.jpg) |
Geometry Problem 1547: Tangents' Dance: Exploring B-to-AC Distance in a
Circle's Grasp |
![Geometry Problem 1546](../school-college/6/p1546-inscribed-circle-given-sides-calculate-contact-triangle-area-1.jpg) |
Geometry Problem 1546: Discover the Hidden Geometry: Calculate Area of Contact Triangle DEF in Triangle ABC |
![Geometry Problem 1545](../school-college/6/p1545-geometric-puzzle-area-of-triangle-inscribed-circle-excenters-incircle-1.jpg) |
Geometry Problem 1545: Unlock the Geometric Mystery: Calculate the Area of Triangle ABC with Inscribed Circle and Excenters! |
![Geometry Problem 1544](../school-college/6/p1544-calculate-area-triangle-with-given-arc-semicircle-intersection-geometry-resource-1.jpg) |
Geometry Problem 1544: Challenge: Calculate the Area of a Triangle with Given Arc and Semicircle Intersections. |
![Geometry Problem 1540](../school-college/6/p1540-length-chord-circle-intersection-diameter-similarity-math-class-1.jpg) |
Geometry Problem 1540: Solving for the Length of Chord in a Circle: Analyzing Intersections and Given Values for Academic Pursuits. |
![Geometry Problem 1539](../school-college/6/p1539-isosceles-triangle-tangent-circle-altitude-right-angle-midpoint-measurement-math-class-1.jpg) |
Geometry Problem 1539 Demystified: Unraveling the Lengths in an Isosceles Triangle with Altitude and Tangent Secrets!. |
![Geometry Problem 1535](../school-college/6/p1535-inscribed-circle-in-square-find-angle-crack-the-code-thematic-poem-1.jpg) |
Geometry
Problem 1535: Crack the Code: Inscribed Circle in Square - Angle Challenge! Solve the Mystery. |
![Geometry Problem 1534](../school-college/6/p1534-tangent-circle-common-external-tangent-angle-brainteaser-1.jpg) |
Geometry
Problem 1534: High School Brainteaser: Tangent Circles, Common External Tangent, and Angle Conundrums!.
. |
![Geometry Problem 1532](../school-college/6/p1532-square-inner-semicircle-tangent-find-angle-chord-1.jpg) |
Geometry
Problem 1532: Crack the Code of Geometry Problem 1532: How to Find the Angle in a Square with a Tangent Semicircle! - A High School Challenge.
. |
![Geometry Problem 1528](../school-college/6/p1528-90-degree-circular-sector-inscribed-circle-tangent-find-angle-radius-1.jpg) |
Geometry
Problem 1528: Cracking the Circle Code: Unveiling the Tangent and Angle of an Inscribed Circle within a 90-Degree Circular Sector.
. |
![Geometry Problem 1503](../school-college/6/p1503-triangle-transversal-line-tangent-incircle-distance-vertices-perpendicular-congruence-1.jpg) |
Geometry
Problem 1503.
Triangle, Incircle, Tangent, Congruence, Perpendicular. |
![Geometry Problem 1502](../school-college/6/p1502-rigth-triangle-incircles-measurement-inradius-square-quadrilateral-1.jpg) |
Geometry
Problem 1502.
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral. |
![Geometry Problem 1500](../school-college/5/p1500-circle-tangent-secant-perpendicular-parallel-similarity-cyclic-quadrilateral-1.jpg) |
Geometry
Problem 1500.
Circle, Perpendicular, Tangent, Secant, Cyclic Quadrilateral, Parallel, Similarity, Measurement. |
![Geometry Problem 1492](../school-college/5/p1492-right-triangle-altitude-incenter-angle-measurement-1.gif) |
Geometry
Problem 1492.
Right Triangle, Altitude, Incenters, Angle, Measurement. |
![Geometry Problem 1491](../school-college/5/p1491-cyclic-quadrilateral-diagonal-incircle-angle-measurement-1.gif) |
Geometry
Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement. |
![Geometry Problem 1490](../school-college/5/p1490-triangle-cevian-incircle-tangent-measurement-1.gif) |
Geometry
Problem 1490.
Right Triangle, Angle Bisectors, Perpendicular, Measurement. |
![Geometry Problem 1486](../school-college/5/p1486-right-tringle-altitude-incircle-tangent-measurement-1.jpg) |
Geometry
Problem 1486.
Right Triangle, Altitude, Incircle, Tangent, Measurement. |
![Geometry Problem 1485](../school-college/5/p1485-triangle-altitude-orthocenter-circle-tangent-measurement-1.jpg) |
Geometry
Problem 1485.
Triangle, Orthocenter, Altitude, Circle, Diameter, Tangent, Measurement. |
![Geometry Problem 1484](../school-college/5/p1484-right-triangle-three-incircles-inradius-altitude-measurement-1.jpg) |
Geometry
Problem 1484.
Right Triangle, Altitude, Incircles, Inradius, Measurement. |
![Dynamic Geometry 1483](../school-college/5/p1483-isosceles-right-triangle-excenter-perpendicular-measurement-1.jpg) |
Geometry
Problem 1483.
Isosceles Right Triangle, Excenter, Perpendicular, Measurement. |
![Dynamic Geometry 1481](../school-college/5/p1481-5-circumscribed-quadrilateral-tangent-pitot-ipad-1.jpg) |
Dynamic Geometry 1481.
Five Tangential or Circumscribed Quadrilaterals, Pitot Theorem, Congruence, Step-by-step Illustration. GeoGebra,
iPad. |
![Dynamic Geometry 1471](../school-college/5/p1471-equilateral-triangle-incenter-concurrent-geogebra-ipad-1.jpg) |
Dynamic Geometry 1471.
Equilateral Triangle, Inside/Outside Point, Incenters, Tangency Points, Concurrent Lines, Step-by-step Illustration. |
![Dynamic Geometry 1470](../school-college/5/p1470-tangential-quadrilateral-circle-rhombus-geogebra-1.jpg) |
Dynamic Geometry 1470.
Tangential Quadrilateral, Incircles, Tangent,
Parallel, Rhombus, Step-by-step Illustration. |
![Dynamic Geometry 1469](../school-college/5/p1469-triangle-inradius-circumradius-midpoint-geogebra-1.jpg) |
Dynamic Geometry 1469.
Triangle, Circumradius, Inradius, Midpoints, Arcs, Sum of Distances, Step-by-step Illustration. |
![Dynamic Geometry 1468](../school-college/5/p1468-steiner-theorem-triangle-inradius-exradius-circumradius-ipad-geogebra-1.jpg) |
Dynamic Geometry 1468.
Steiner's Theorem, Triangle, Circumradius, Inradius, Sum of Exradii, Step-by-step Illustration. |
![Geometry Problem 1466](../school-college/5/p1466-tangential-quadrilateral-newton-line-incenter-geogebra-ipad-1.jpg) |
Geometry Problem 1466.
Tangential Quadrilateral, Newton Line, Incenter, Midpoint, Diagonal, Collinear Points.
Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1465](../school-college/5/p1465-tangential-quadrilateral-incenter-equal-sum-area-1.jpg) |
Geometry Problem 1465.
Tangential Quadrilateral, Incenter, Inscribed Circle, Equal Sum of Areas.
Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1458](../school-college/5/p1458-incircle-excircle-triangle-area-1.jpg) |
Geometry
Problem 1458.
Triangle, Incircles, Excircle, Area, Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1455](../school-college/5/p1455-nagel-point-incircle-congruence-geogebra-ipad-1.jpg) |
Geometry
Problem 1455.
Nagel Point, Excircles, Incircle, Congruent Segments,
iPad. Step-by-step illustration using GeoGebra. |
![Problem 1446](../school-college/5/p1446-lemoine-line-tangent-geogebra-dynamic-1.jpg) |
Dynamic Geometry Problem 1446.
Lemoine Line, triangle, circumcircle, tangent, collinear points. Step-by-step animation using GeoGebra. |
![Problem 1437](../school-college/5/p1437-circle-intersecting-tangent-equal-product-ipad-sw-typography-1.jpg) |
Geometry Problem 1437.
Intersecting Circles, Common Tangent, Equal Product, Art, Poster, Typography, iPad Apps. |
![Problem 1436](../school-college/5/p1436-right-triangle-altitude-excircle-geometric-mean-1.jpg) |
Geometry Problem 1436.
Right Triangle, Altitude, Excircles, Excenters, Geometric Mean,
Measurement, Art,
Poster, Typography, iPad Apps. |
![Problem 1435](../school-college/5/p1435-circle-diameter-tangent-parallelogram-1.jpg) |
Geometry Problem 1435.
Circle, Diameter, Inscribed Circles, Circular Sector, Parallelogram, Parallel Lines, Tangency Points,
Poster, Typography, iPad Apps. |
![Problem 1434](../school-college/5/p1434-equilateral-triangle-tangent-circles-line-1.jpg) |
Geometry Problem 1434.
Equilateral Triangle, Tangent Circles, Tangent Lines, Arithmetic Mean, Measurement,
Poster, Typography, iPad Apps. |
![Problem 1433](../school-college/5/p1433-tangent-circles-line-midpoint-1.jpg) |
Geometry Problem 1433.
Tangent Circles, Diameter, Perpendicular, Midpoint, Measurement,
Poster Typography, iPad Apps. |
![Problem 1432](../school-college/5/p1432-tangent-circles-proportion-similarity-ipad-apps-1.jpg) |
Geometry Problem 1432.
Tangent Circles, Secant, Tangent Lines, Proportionality, Similarity,
Poster Typography, iPad. |
![Problem 1431](../school-college/5/p1431-apollonius-llc-triangle-circumcircle-tangent-congruent-angle-1.jpg) |
Geometry Problem 1431.
Triangle, Circumcircle, Parallel Chord, Apollonius, LLC, Congruent Angles, Poster, Conformal Mapping. |
![Problem 1428](../school-college/5/p1428-circle-tangent-triangle-square-area-1.jpg) |
Geometry Problem 1428. Intersecting Circles, Tangent Line, Triangle, Square, Area. |
![Problem 1427](../school-college/5/p1427-square-circle-tangent-radius-1.jpg) |
Geometry Problem 1427. Circle inscribed in a square, Arc, Tangent Line, Tangent Circles, Radii. |
![Problem 1421](../school-college/5/p1421-right-triangle-incircle-excircle-measurement-1.jpg) |
Geometry Problem 1421. Right Triangle, Incircle, Excircle, Tangent Lines, Measurement. |
![Problem 1420](../school-college/5/p1420-circle-area-regular-hexagon-tangent-1.jpg) |
Geometry Problem 1420. Regular Hexagon, Inscribed Circle, Area, Tangent
Circles, Tangent Line. |
![Problem 1419](../school-college/5/p1419-triangle-incenter-inradius-circumcircle-distance-sketch-ipad-1.jpg) |
Geometry Problem 1419. Triangle, Altitude, Circumcircle, Incenter, Inradius, Perpendicular, Angle Bisector, Midpoint. |
![Problem 1418](../school-college/5/p1418-triangle-incenter-parallel-angle-circle-tangent-1.jpg) |
Geometry Problem 1418. Triangle, Circumcircle, Incenter, Side, Sum of Segments, Tangent, Parallel, Angle. |
![Problem 1417](../school-college/5/p1417-triangle-incenter-circumcircle-tangent-cevian-bisector-perpendicular-1.jpg) |
Geometry Problem 1417. Triangle, Cevian, Circumcircle, Incenter, Perpendicular, Angle Bisector,
Tangent Lines, Tangent Circles. |
![Problem 1416](../school-college/5/p1416-right-triangle-altitude-incircle-excircle-tangent-45-degree-1.jpg) |
Geometry Problem 1416. Right Triangle, Altitude, Incircle, Excircle, Tangency Points, 45 Degree Angle. |
![Problem 1415](../school-college/5/p1415-right-triangle-altitude-incircle-excircle-tangent-isosceles-1.jpg) |
Geometry Problem 1415. Right Triangle, Altitude, Incircle, Excircle, Tangency Points, Isosceles Triangle. |
![Problem 1414](../school-college/5/p1414-right-triangle-altitude-incircle-excircle-tangent-isosceles-1.jpg) |
Geometry Problem 1414. Right Triangle, Altitude, Incircle, Excircle, Tangency Points, Isosceles Triangle. |
![Problem 1413](../school-college/5/p1413-right-triangle-incircle-excircle-tangent-isosceles-1.jpg) |
Geometry Problem 1413. Right Triangle, Incircle, Excircle, Tangency Points,
Isosceles Right Triangle. |
![Problem 1412](../school-college/5/p1412-right-triangle-incircle-excircle-tangent-45-degrees-1.jpg) |
Geometry Problem 1412. Right Triangle, Incircle, Excircle, Tangency Points,
45 Degree Angle. |
![Problem 1411](../school-college/5/p1411-right-triangle-incircle-excircle-tangent-isosceles-1.jpg) |
Geometry Problem 1411. Right Triangle, Incircle, Excircle, Tangency Points,
Isosceles Right Triangle. |
![Problem 1410](../school-college/5/p1410-right-triangle-incircle-excircle-tangent-45-degrees-1.jpg) |
Geometry Problem 1410. Right Triangle, Incircle, Excircle, Tangency Points,
45 Degree Angle. |
![Problem 1409](../school-college/5/p1409-right-triangle-incircle-excircle-tangent-collinear-1.jpg) |
Geometry Problem 1409. Right Triangle, Incircle, Excircle, Collinear Tangency Points, Collinearity. |
![Problem 1408](../school-college/5/p1408-right-triangle-incircle-excircle-midpoint-collinear-1.jpg) |
Geometry Problem 1408. Right Triangle, Incircle, Excircle, Incenter, Midpoint, Tangency Point, Collinearity. |
![Problem 1407](../school-college/5/p1407-right-triangle-incircle-excircle-collinearity-1.jpg) |
Geometry Problem 1407. Right Triangle, Incircle, Excircle, Collinear Tangency Points, Collinearity. |
![Problem 1401](../school-college/5/p1401-rigth-triangle-isosceles-tangent-congruence-tutor-1.jpg) |
Geometry Problem 1401. Right Triangle with three circles on the sides, Isosceles, Diameter, Center, Tangent, Congruence. |
![Problem 1398](../school-college/4/p1398-isosceles-triangle-circle-measurement-tutor-1.jpg) |
Geometry Problem 1398.
Isosceles Triangle, Circle, Radius, Perpendicular, Measurement. |
![Problem 1397](../school-college/4/p1397-circle-tangent-90-degree-collinear-tutoring-1.jpg) |
Geometry Problem 1397. Circle, Diameter, Tangent, Secant, 90 Degrees Angle,
Perpendicular, Collinear Points. |
![Problem 1394](../school-college/4/p1394-three-circles-common-internal-tangent-concurrent-1.jpg) |
Geometry Problem 1394. Three Circles and Three Pair of Common Internal Tangents, Concurrency. |
![Problem 1393](../school-college/4/p1393-monge-circle-theorem-tangent-collinear-tutor-1.jpg) |
Geometry Problem 1393. Monge's Circle Theorem, Three Circles and Three Pair of Common Tangents, Collinearity. |
![Problem 1380 Circles, Tangents, Chords](../school-college/4/p1380-two-circle-common-internal-tangent-congruent-chord-1.jpg) |
Geometry Problem 1380. Common
Internal Tangents. Circles, Secant, Congruent Chords. |
![Problem 1379 Circles, Tangents, Chords](../school-college/4/p1379-two-circle-common-external-tangent-congruent-chord-1.jpg) |
Geometry Problem 1379. Common External Tangents. Circles, Secant, Congruent Chords. |
![Geometry problem 1362](../school-college/4/p1362-equilateral-triangle-inscribed-circle-lunula-area-1.jpg) |
Geometry Problem 1362.
Equilateral Triangle Inscribed in a Circle, Lunula, Ratio of Areas, Tangent, Sketch, iPad Apps. |
![Geometry problem 1365](../school-college/4/p1365-circle-chord-parallel-secant-tangent-midpoint-1.jpg) |
Geometry Problem 1365.
Circle, Tangent, Secant, Chord, Parallel, Midpoint, Sketch, iPad Apps. |
![Geometry problem 1354](../school-college/4/p1354-circle-tangent-secante-parallel-tutoring-1.jpg) |
Geometry Problem 1354. Circle, Tangent Lines, Secant, Chord, Center, Parallel Lines. |
![Geometry problem 1347](../school-college/4/p1347-equilateral-triangle-bisector-circle-tangent-tutoring-1.jpg) |
Geometry Problem 1347. Triangle, Angle Bisector, Congruence, Circumcircle, Tangent, Equilateral Triangle. |
![Geometry problem 1343](../school-college/4/p1343-triangle-excircle-tangent-90-degree-1.jpg) |
Problem 1343.
Triangle, Excircles, Circle, Tangent, Tangency Points, Chord, Perpendicular, 90 Degrees, Collinearity. Machu Picchu in the background.
|
![Geometry problem 1340](../school-college/4/p1340-scalene-triangle-incenter-circle-isosceles-1.jpg) |
Problem 1340.
Triangle, Incenter, Concentric Circles, Isosceles Triangles, Congruence.
|
![Geometry problem 1339](../school-college/4/p1339-circle-diameter-tangent-secant-parallelogram-1.jpg) |
Problem 1339.
Circle, Diameter, Tangent, Secant, Parallelogram.
|
![Geometry problem 1337](../school-college/4/p1337-parallelogram-circumcircle-tangent-congruence-tutoring-1.jpg) |
Problem 1337.
Parallelogram, Diagonals, Circle, Circumcircle, Tangent Line, Triangle, Congruence.
|
![Geometry problem 1320](../school-college/4/p1320-triangle-incircle-tangent-chord-collinear-1.jpg) |
Geometry Problem 1320.
Triangle, Incircle, Tangent, Chord, Circle, Parallel, Perpendicular, Collinearity. |
![Geometry problem 1317](../school-college/4/p1317-triangle-excircle-tangent-chord-congruence-1.jpg) |
Geometry Problem 1317.
Triangle, Excircle, Chord, Tangent, Midpoint, Arc, Sum of two Segments, Congruence. |
![Geometry problem 1316](../school-college/4/p1316-triangle-incircle-chord-sum-segment-congruence-1.jpg) |
Geometry Problem 1316.
Triangle, Incircle, Tangent, Chord, Midpoint, Sum of two Segments, Congruence. |
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Last updated: Jul 26, 2024
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