The figure below shows a point P inside (or
outside) an equilateral triangle ABC. The points O_{1}, O_{2}, and O_{3} are the incenters of triangles APB, BPC and APC so that T_{1}, T_{2}, and T_{3} are tangency points. Prove that the lines T_{1}O_{1}, T_{2}O_{2}, and T_{3}O_{3} are concurrent
at a point D.

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

Triangle Centers

Incenter, Incircle

Tangency Point

Perpendicular lines

Concurrent lines

Dynamic Geometry

GeoGebra

HTML5 and Dynamic Geometry

iPad Apps

View or Post a solution