Challenge Your Geometry Skills with Problem 1456: Altitudes, Orthic Triangle, Circumcircle, Parallel, Similarity, and Area

Let AHA, BHB, CHC be the altitudes of triangle ABC. The extensions of AHA, BHB, and CHC intersect the circumcircle O at A1, B1, and C1, respectively. Prove that (1) HAHB // A1B1, HBHC // B1C1, and HAHC // A1C1; (2) the area of triangle A1B1C1 is 4 times the area of triangle HAHBHC.

See solution below

Static Diagram of problem 1456

Dynamic Geometry 1456: Altitude, Orthic Triangle, Circumcircle, Similarity, Area. Using GeoGebra

Poster of the problem 1456 using iPad Apps

Poster of Problem 1456, Altitude, Orthic Triangle, Circumcircle, Similarity, Area. Using iPad


Classroom Resource:
Interactive step-by-step animation using GeoGebra

This step-by-step interactive illustration was created with GeoGebra.

  • To explore (show / hide): click/tap a check box.
  • To stop/play the animation: click/tap the icon in the lower left corner.
  • To go to first step: click/tap the "Go to step 1" button.
  • To manipulate the interactive figure: click/tap and drag the blue points or figures.

GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus application, intended for teachers and students. Many parts of GeoGebra have been ported to HTML5.

Recent Additions

Geometry Problems
Open Problems
Visual Index
Ten problems: 1411-1420
All Problems
Triangle Centers
Perpendicular lines
Parallel lines
Orthic Triangle
Similarity, Ratios, Proportions
Triangle Area
Dynamic Geometry
HTML5 and Dynamic Geometry
iPad Apps
View or Post a solution

Geometry Problem 1456 Solution(s)