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![College Geometry Online, eLearning](../machu_picchu_h1_810.jpg)
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Altitude of a Triangle, Theorems and Problems - Table of Content 1
Altitude, so high,
Triangle's perpendicular,
Theorems come alive.
Problems unravel with ease,
Altitude unveils secrets.
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![Altitude of a triangle](altitude_triangle_17.jpg) |
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![Euclid Elements Book X, Lemma for Proposition 33](euclid_book_x_33_17.jpg) |
Euclid's
Elements, Book X, Lemma for Proposition 33
One page visual illustration
Altitude. |
![Orthocenter Index](../center/orthocenter_altitude_index.jpg) |
Orthocenter
Index.
Altitudes.
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![Orthic Triangle Index](orthic_triangle_index_1.jpg) |
Orthic Triangle
Index.
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![Euclid's Elements, Book II, Proposition 12](euclid_elements_ii_12_17.jpg) |
Euclid's Elements Book II, Proposition 12: Law of Cosines.
Altitude.
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![Euclid's Elements: Book II, Proposition 13: Law of Cosines](euclid_elements_ii_13_17.jpg) |
Euclid's Elements Book II, Proposition 13: Law of Cosines.
Altitude.
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![Right Triangle, Altitude, Incircle Index](../geometry/right-triangle-altitude-bisector-incircle-inradius-theorems-problems-i.jpg) |
Right triangle, Altitude, Incircle
Right Triangle, Altitude to the Hypotenuse, Incircle, Incenter, Inradius, Angle Bisector, Theorems and Problems, Index. |
![Dynamic Geometry: Orthocenter of a Triangle. HTML5 Animation for Tablets (iPad, Nexus..)](../geogebra/geogebra-triangle-orthocenter-html5-animation-ipad-nexus-1.jpg) |
GeoGebra, Dynamic Geometry:
Orthocenter
of a Triangle.
HTML5 Animation for Tablets (iPad, Nexus..) |
![Dynamic Geometry: Nine Point Circle. HTML5 Animation for Tablets (iPad, Nexus..)](../geogebra/nine-point-circle-euler-feuerbach-html5-animation-ipad-nexus-1.jpg) |
GeoGebra, Dynamic Geometry:
Nine Point Circle
HTML5 Animation for Tablets (iPad, Nexus..) |
![Geometry Problem 1563](../school-college/6/p1563-midpoint-mysteries-proving-perpendicularity-in-right-triangle-abc-altitude-bisectors-1.jpg) |
Geometry Problem 1563:
Perpendicularity in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. |
![Geometry Problem 1562](../school-college/6/p1562-right-angled-triangle-bisector-midpoint-collinear-points-proof-1.jpg) |
Geometry Problem 1562:
Proof of Collinearity in a Right-angled Triangle involving the Altitude, Angle bisectors, and Midpoint. |
![Geometry Problem 1561](../school-college/6/p1561-highschool-college-geometry-triangle-ABC-circumcenter-orthocenter-angle-parallel-problem-1.jpg) |
Geometry Problem 1561:
Triangle ABC, Circumcenter O, Orthocenter H, Parallel Line, and Angle
Secrets. |
![Dynamic Geometry: Internal Feuerbach Point. HTML5 Animation for Tablets iPad, Nexus](../geogebra/internal-feuerbach-point-html5-animation-tablet-ipad-nexus-1.jpg) |
GeoGebra, Dynamic Geometry:
Internal Feuerbach Point
HTML5 Animation for Tablets (iPad, Nexus..) |
![Geometry Problem 1554](../school-college/6/p1554-triangle-ABC-angles-A-B-altitude-AH-median-AM-length-HM-4-units-length-AB-1.jpg) |
Geometry Problem 1554: Finding the Length of Side AB in Triangle ABC |
![Geometry Problem 1548](../school-college/6/p1548-triangle-abc-angle-a-45-altitudes-bd-ce-intersect-point-f-de-length-calculation-1.jpg) |
Geometry Problem 1548: Exploring Segment DE Length in Triangle ABC with a 45-Degree Angle and Intersecting Altitudes |
![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 1523](../school-college/6/p1523-calculate-length-altitude-isosceles-triangle-midpoint-1.jpg) |
Geometry
Problem 1523 and a Thematic Poem.
Discover How to Calculate the Length of the Altitude in an Isosceles Triangle - Get Expert Geometry Tips Now!
. |
![Geometry Problem 1507](../school-college/6/p1507-discover-isosceles-triangle-doble-angle-measurement-altitude-perpendicular-1.jpg) |
Geometry
Problem 1507.
Understanding Triangle Geometry: Double Angles, Altitudes, and Measurements. Difficulty Level: High School. |
![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 1488](../school-college/5/p1488-right-triangle-altitude-angle-bisector-measurement-1.gif) |
Geometry
Problem 1488.
Right Triangle, Altitude, Incircle, Tangent, Measurement. |
![Geometry Problem 1487](../school-college/5/p1487-right-triangle-altitude-angle-bisector-measurement-1.gif) |
Geometry
Problem 1487.
Right Triangle, Altitude, Incircle, Tangent, 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 1476](../school-college/5/p1476-droz-farny-line-theorem-orthocenter-collinear-midpoint-geogebra-1.jpg) |
Dynamic Geometry 1476.
Droz-Farny Line Theorem, Triangle, Orthocenter, Perpendicular, Collinear Midpoints, Step-by-step Illustration. GeoGebra,
iPad. |
![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 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. |
![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 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 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 1375 Isosceles Triangle, Circle](../school-college/4/p1375-isosceles-triangle-interior-cevian-exradius-altitude-1.jpg) |
Geometry Problem 1375. Isosceles Triangle, Interior Cevian, Exradius, Excircle, Altitude to the Base. |
![Problem 1373 Isosceles Triangle, Altitude](../school-college/4/p1373-equilateral-triangle-interior-cevian-inradius-exradius-altitude-1.jpg) |
Geometry Problem 1373. Isosceles Triangle, Exterior Cevian, Inradius, Exradius, Altitude to the Base. |
![Geometry problem 1346](../school-college/4/p1346-equilateral-triangle-incircle-altitude-tutoring-1.jpg) |
Geometry Problem 1346. Equilateral Triangle, Point on the Incircle, Altitude, Perpendicular, Sum of Squares, Distance. |
![Geometry problem 1312](../school-college/4/p1312-triangle-60-degrees-circumcenter-orthocenter-measurement-1.jpg) |
Geometry Problem 1312.
Triangle, 60 Degrees, Circle, Circumcenter, Orthocenter, Altitude, Distance, Perpendicular, Measurement. |
![Geometry problem 1294](../school-college/3/p1294-triangle-square-altitude-circle-geometry-1.jpg) |
Geometry Problem 1294.
Triangle, Altitudes, Four Squares, Center, Midpoint, Concyclic Points. |
![Geometry problem 1293](../school-college/3/p1293-triangle-square-altitude-concurrent-1.jpg) |
Geometry Problem 1293.
Triangle, Altitudes, Two Squares, Center, Concurrent Lines, Midpoint. |
![Geometry problem 1281](../school-college/3/p1281-triangle-circle-diameter-tangent-angle-bisector-tutoring-1.jpg) |
Geometry Problem 1281.
Triangle, Circle, Diameter, Tangent, Altitude, Congruent Angles. |
![Geometry problem 1272](../school-college/3/p1272-isosceles-triangle-altitude-midpoint-angle-1.jpg) |
Geometry Problem 1272.
Isosceles Triangle, Median, Midpoint, Perpendicular, 90 Degrees, Angles, Congruence. |
![Geometry problem 1262](../school-college/3/p1262-isosceles-triangle-altitude-median-90-degree-1.jpg) |
Geometry Problem 1262.
Isosceles Triangle, Altitude, Medians, 90 Degrees, Perpendicular. |
![Geometry problem 1257](../school-college/3/p1257-triangle-altitude-90-degree-measurement-1.jpg) |
Geometry Problem 1257.
Triangle, Altitude, Perpendicular, 90 Degree, Measurement. |
![Geometry problem 1253](../school-college/3/p1253-triangle-orthocenter-90-degree-circles-1.jpg) |
Geometry Problem 1253.
Triangle, Orthocenter, Circle, Circumcircle, Altitude, Perpendicular, 90 Degree, Concurrency. |
![Geometry problem 1252](../school-college/3/p1252-hexagon-triangle-area-orthocenter-circumcircle-mind-map-1.jpg) |
Geometry Problem 1252.
Triangle, Circle, Circumcircle, Altitude, Area, Hexagon, Perpendicular, 90 Degrees. Mind Map. |
![Geometry problem 1248](../school-college/3/p1248-triangle-circle-orthocenter-diameter-parallel-1.jpg) |
Geometry Problem 1248.
Triangle, Circumcircle, Circle, Diameter, Orthocenter, Altitude, Parallel Lines. |
![Geometry problem 1247](../school-college/3/p1247-triangle-circumcircle-perpendicular-orthocenter-1.jpg) |
Geometry Problem 1247.
Triangle, Circumcircle, Orthocenter, Altitude, Perpendicular, Midpoint. |
![Geometry problem 1246](../school-college/3/p1246-orthocenter-midpoint-perpendicular-1.jpg) |
Geometry Problem 1246.
Triangle, Orthocenter, Altitude, Midpoint, Perpendicular. |
![Geometry problem 1233](../school-college/3/p1233-orthic-axis-triangle-euler-line-perpendicular-1.jpg) |
Geometry Problem 1233
Triangle, Euler Line, Orthic Axis, Perpendicular, 90 Degrees, Orthic Triangle. |
![Geometry problem 1232](../school-college/3/p1232-orthic-axis-triangle-altitude-collinear-1.jpg) |
Geometry Problem 1232
Triangle, Altitudes, Orthic Triangle, Collinearity, Orthic Axis. |
![Geometry problem 1213](../school-college/3/p1213-triangle-altitudes-circumcircle-cyclic-quadrilateral-1.jpg) |
Geometry Problem 1213
Triangle, Circumcircle, Altitudes, Cyclic Quadrilateral, Congruence. |
![Geometry problem 1211](../school-college/3/p1211-right-triangle-altitude-bisector-45-degrees-class-1.jpg) |
Geometry Problem 1211
Right Triangle, Altitude, Angle Bisector, 45 Degrees. |
![Geometry problem 1199](../school-college/2/p1199-square-equilateral-triangle-circle-metric-relation-1.jpg) |
Geometry Problem 1199
Equilateral Triangle, Square, Altitude, Circle, Incircle, Inradius, Metric Relations. |
![Geometry problem 1171](../school-college/2/p1171-area-a-triangle-altitude-height-ha-hb-hc-terms-proof-1.jpg) |
Geometry Problem 1171
Area of a Triangle in terms of the three altitude or
heights ha, hb, hc. |
![Geometry problem 1170](../school-college/2/p1170-area-of-a-triangle-three-sides-a-b-c-proof-1.jpg) |
Geometry Problem 1170
Area of a Triangle in terms of the three sides, a-b-c. |
![Geometry problem 1169](../school-college/2/p1169-ortholine-complete-quadrilateral-steiner-line-tutoring-1.jpg) |
Geometry Problem 1169
Complete Quadrilateral, Orthocenters of the Component Triangles, Collinear Points, Ortholine, Steiner-line, Orthocentric line. |
![Geometry problem 1155](../school-college/2/p1155-right-triangle-altitude-incenters-area-1.jpg) |
Geometry Problem 1155.
Right Triangle, Altitude, Incenter, Incircle, Area. |
![Geometry problem 1150](../school-college/2/p1150-circle-secant-90-degrees-perpendicular-congruence-1.jpg) |
Geometry Problem 1150.
Circle, Perpendicular Secants, 90 Degrees, Congruence. |
![Geometry problem 1149](../school-college/2/p1149-isosceles-triangle-cevian-incircles-tangent-congruent-1.jpg) |
Geometry Problem 1149.
Isosceles Triangle, Altitude, Cevian, Incircles, Tangent, Congruence. |
![Geometry Problem 1073](../school-college/1/p1073-triangle-altitude-exterior-point-distance-sides-ratio-geometry-math-1.jpg) |
Geometry Problem
1073.
Triangle, Altitudes, Exterior Point, Distance, Ratio. |
![Geometry Problem 1072](../school-college/1/p1072-triangle-altitude-point-distance-sides-ratio-geometry-math-1.jpg) |
Geometry Problem
1072.
Triangle, Altitudes, Interior Point, Distance, Ratio. |
![Geometry Problem 1068](../school-college/1/p1068-obtuse-triangle-orthocenter-exradius-inradius-circumradius-distance-math-1.jpg) |
Geometry Problem
1068.
Obtuse Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter |
![Geometry Problem 1067](../school-college/1/p1067-triangle-orthocenter-exradius-inradius-circumradius-distance-math-1.jpg) |
Geometry Problem
1067.
Acute Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter. |
![Geometry Problem 1066](../school-college/1/p1066-triangle-obtuse-orthocenter-inradius-circumradius-distance-math-1.jpg) |
Geometry Problem
1066.
Triangle, Obtuse Angle, Orthocenter, Circumradius R, Inradius r, Exradius
ra, Distance, Diameter. |
![Geometry Problem 1065](../school-college/1/p1065-triangle-orthocenter-inradius-circumradius-distance-math-1.jpg) |
Geometry Problem
1065.
Triangle, Acute Angle, Orthocenter, Circumradius R, Inradius r, Exradius
ra, Distance, Diameter. |
![Geometry Problem 1064](../school-college/1/p1064-triangle-orthocenter-sides-circumradius-square-math-1.jpg) |
Geometry Problem
1064.
Triangle, Orthocenter, Altitudes, Circumradius, Sum of the Squares of Sides. |
![Geometry Problem 1063](../school-college/1/p1063-triangle-altitudes-orthocenter-equal-product-math-1.jpg) |
Geometry Problem
1063.
Triangle, Orthocenter, Altitudes, Equal Product of the Lengths of Segments. |
![Geometry Problem 1059](../school-college/1/p1059-triangle-circle-simson-line-reflection-parallel-perpendicular-math-1.jpg) |
Geometry Problem
1059.
Triangle, Circumcircle, Orthocenter, Altitude, Perpendicular, Reflection of a Point over a Line, Collinear Points, Simson Line, Congruence. |
![Geometry Problem 1058](../school-college/1/p1058-triangle-sum-area-altitude-circle-diameter-perpendicular-math-1.jpg) |
Geometry Problem
1058.
Triangle, Sum of the Areas, Altitude, Perpendicular, Circumcircle, Circle, Diameter. |
![Geometry Problem 1056](../school-college/1/p1056-triangle-reciprocals-exradius-altitudes-formula-exradii-math-1.jpg) |
Geometry Problem
1056.
Triangle, Exradius, Reciprocals of the Altitudes, Multiplicative Inverse, Perpendicular, Excircle, Circle. |
![Geometry Problem 1055](../school-college/1/p1055-triangle-area-altitude-circumradius-perpendicular-math-1.jpg) |
Geometry Problem
1055.
Triangle, Altitude, Perpendicular, Area, Circumradius, Circle. |
![Geometry Problem 1054](../school-college/1/p1054-triangle-circle-altitude-midpoint-orthocenter-circumcenter-math-1.jpg) |
Geometry Problem
1054.
Triangle, Altitude, Orthocenter, Circumcenter, Perpendicular, Midpoints. |
![Geometry Problem 1016](../school-college/1/p1016-triangle-45-degrees-angle-altitude-bisector-quadrilateral-area-math-1.jpg) |
Geometry Problem
1016.
Triangle, 45 Degrees, Area, Triangle, Quadrilateral, Altitude, Bisector. |
![Geometry Problem 1015](../school-college/1/p1015-right-triangle-square-altitude-metric-relations-math-1.jpg) |
Geometry Problem
1015.
Right Triangle, Square, Altitude, Metric Relations. |
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