Let AH_{A}, BH_{B},
CH_{C}
be the altitudes of triangle ABC. The extensions of AH_{A}, BH_{B},
and CH_{C}
intersect the circumcircle O at A_{1}, B_{1}, and C_{1,
}respectively.
Prove that (1) H_{A}H_{B} // A_{1}B_{1},
H_{B}H_{C} // B_{1}C_{1}, and H_{A}H_{C} // A_{1}C_{1};
(2) the area of triangle A_{1}B_{1}C_{1} is 4 times the area of
triangle H_{A}H_{B}H_{C}.

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.

Geometry Problems

Open Problems

Visual Index

Ten problems: 1411-1420

All Problems

Triangle

Circle

Circumcircle

Triangle Centers

Altitude

Orthocenter

Perpendicular lines

Parallel lines

Orthic Triangle

Similarity, Ratios, Proportions

Triangle Area

Dynamic Geometry

GeoGebra

HTML5 and Dynamic Geometry

iPad Apps

View or Post a solution