Triangles: Theorems and Problems 4
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![Triangles](../geometry/triangles_index_17.jpg) |
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![Geometry Problem 1461](../school-college/5/p1462-newton-theorem-gauss-line-quadrilateral-midpoint-side-diagonal-geogebra-1.jpg) |
Geometry Problem 1462.
Newton-Line, Newton-Gauss Line, Complete Quadrilateral, Midpoints of Sides and Diagonals,
Collinear Points. |
![Geometry Problem 1461](../school-college/5/p1461-quadrilateral-angle-30-congruence-1.jpg) |
Geometry Problem 1461.
Quadrilateral, Triangle, Angles, 30-60 Degree, Congruence, Auxiliary Lines. |
![Geometry Problem 1460](../school-college/5/p1460-newton-gauss-line-dynamic-complete-quadrilateral-1.jpg) |
Dynamic Geometry
1460.
Newton-Gauss Line, Complete Quadrilateral, Midpoints of Diagonals,
Collinear Points, Step-by-step Illustration using GeoGebra. |
![Geometry Problem 1459](../school-college/5/p1459-orthocenter-triangle-midpoint-perpendicular-1.jpg) |
Geometry
Problem 1459.
Two Triangles, Orthocenter, Midpoint, Perpendicular, 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 1457](../school-college/5/p1457-orthocenter-triangle-similarity-circles-radii-1.jpg) |
Geometry
Problem 1457.
Altitudes, Circles, Similarity,
Product of the Inradii Lengths,
iPad. Step-by-step illustration using GeoGebra. |
![Geometry Problem 1456](../school-college/5/p1456-orthocenter-triangle-similarity-circles-geogebra-ipad-1.jpg) |
Geometry
Problem 1456.
Altitudes, Orthic Triangle, Circumcircle, Parallel lines, Similarity, Area,
iPad. 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. |
![Dynamic Geometry 1452](../school-college/5/p1452-cyclic-quadrilateral-sangaku-japanese-theorem-geogebra-1.jpg) |
Dynamic Geometry 1452.
Japanese Theorem, Sangaku,
Cyclic Quadrilateral, Incenter, Rectangle, Inradius. Step-by-step
illustration using GeoGebra. |
![Dynamic Geometry 1451](../school-college/5/p1451-orthopole-geogebra-triangle-perpendicular-concurrency-1.jpg) |
Dynamic Geometry 1451.
Orthopole of a Line. Step-by-step
illustration using GeoGebra. |
![Dynamic Geometry 1450](../school-college/5/p1450-ortholine-complete-quadrilateral-geogebra-dynamic-igs-1.jpg) |
Dynamic Geometry 1450.
Ortholine, Steiner Line, Complete Quadrilateral, Collineal Orthocenters. Step-by-step
illustration using GeoGebra. |
![Dynamic Geometry 1448](../school-college/5/p1448-simson-line-geogebra-dynamic-1.jpg) |
Dynamic Geometry 1448.
Simson Line. Step-by-step
illustration using GeoGebra. |
![Problem 1447](../school-college/5/p1447-vecten-point-outer-geogebra-dynamic-1.jpg) |
Dynamic Geometry Problem 1447.
Outer Vecten Point. 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 1445](../school-college/5/p1445-van-aubel-theorem-quadrilateral-geogebra-step-1.jpg) |
Dynamic Geometry Problem 1445.
Van Aubel's theorem, Quadrilateral and Four Squares, Centers. Step-by-step animation using GeoGebra. |
![Problem 1444](../school-college/5/p1444-asymmetric-propeller-equilateral-triangle-geogebra-1.jpg) |
Dynamic Geometry Problem 1444.
The Asymmetric Propeller Theorem, Equilateral Triangles, Midpoints. Using GeoGebra. |
![Problem 1443](../school-college/5/p1443-triangle-area-incenter-90-degree-circumcenter-1.jpg) |
Geometry Problem 1443.
Triangle Area, Incenter, Circumcenter, 90 Degree, Perpendicular, One-third
Area, Art, Poster, Typography, iPad Apps. |
![Problem 1442](../school-college/5/p1442-triangle-altitude-angle-measurement-1.jpg) |
Geometry Problem 1442.
Triangle, Altitudes, Orthocenter, Perpendicular, Congruent Angles,
Similarity, Measurement, Art, Poster, Typography, iPad Apps. |
![Problem 1439](../school-college/5/p1439-rectangle-diagonal-perpendicular-circle-ipad-apps-1.jpg) |
Geometry Problem 1439.
Rectangle, Diagonal, Perpendicular, Circles, Areas, Art, Poster, Typography, iPad Apps. |
![Problem 1438](../school-college/5/p1438-triangle-incenter-circuncenter-centroid-parallel-90-degree-ipad-apps-1.jpg) |
Geometry Problem 1438.
Triangle, Incenter, Centroid, Circumcenter, Parallel, 90 Degree, 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 1429](../school-college/5/p1429-triangle-median-angle-15-30-degree-1.jpg) |
Geometry Problem 1429. Triangle, Median, 15-30 degree, Angle, Congruence. |
![Problem 1426](../school-college/5/p1426-two-squares-triangle-quadrilateral-area-1.jpg) |
Geometry Problem 1426. Two Squares, Collinear Points, Triangle, Quadrilateral, Sum of the Areas. |
![Problem 1425](../school-college/5/p1425-two-squares-concurrent-lines-1.jpg) |
Geometry Problem 1425. Two Squares, Collinear Points, Concurrent Lines. |
![Problem 1424](../school-college/5/p1424-parallelogram-triangle-trapezoid-area-1.jpg) |
Geometry Problem 1424. Triangle, Parallelogram, Trapezoid, Area, Diagonal, Measurement. |
![Problem 1423](../school-college/5/p1423-triangle-equal-angle-similarity-area-1.jpg) |
Geometry Problem 1423. Triangle, Area, Three Equal Angles, Circumradius, Measurement. |
![Problem 1422](../school-college/5/p1422-triangle-cevian-transversal-area-1.jpg) |
Geometry Problem 1422. Triangle, Transversal, Cevian, Area, Measurement. |
![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 1402](../school-college/5/p1402-triangle-three-tangent-circles-diameter-collinearity-tutor-1.jpg) |
Geometry Problem 1402. Triangle, Three Tangent Circles, Parallel Diameter, Collinear Points. |
![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 1400](../school-college/4/p1400-triangle-rectangles-perpendicular-bisector-concurrency-1.jpg) |
Geometry Problem 1400. Triangle with three rectangles on the sides, Perpendicular Bisectors, Concurrency. |
![Problem 1399](../school-college/4/p1399-triangle-circumcircle-perpendicular-angle-congruence-1.jpg) |
Geometry Problem 1399.
Triangle, Circumcircle, Perpendiculars, Angles, Congruence, Cyclic Quadrilateral. |
![Problem 1396](../school-college/4/p1396-triangle-rectangles-perpendicular-concurrency-1.jpg) |
Geometry Problem 1396. Triangle with three rectangles on the sides, Vertices, Perpendicular lines, Concurrency. |
![Problem 1395](../school-college/4/p1395-triangle-rectangles-midpoints-concurrency-1.jpg) |
Geometry Problem 1395. Triangle with three rectangles on the sides, Midpoints, Perpendicular lines, Concurrency. |
![Problem 1391](../school-college/4/p1391-triangle-ratio-area-circumcircle-1.jpg) |
Geometry Problem 1391. Two Triangles, Ratio of Areas, Circumcircle. |
![Problem 1390](../school-college/4/p1390-triangle-point-perpendicular-sum-square-tutoring-1.jpg) |
Geometry Problem 1390. Triangle, any point, perpendicular, sum of the squares. |
![Problem 1389 40-120-20 triangle](../school-college/4/p1389-triangle-40-20-degree-angle-congruence-tutoring-1.jpg) |
Geometry Problem 1389. Triangle, 40-120-20 degree, Angle, Congruence. |
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