Saltire Software

Area of the Triangle Formed by the
Intersection of the Tangents of 3 Circles

It is a well known theorem that the exterior common tangents to 3 circles intersect on a line.

The intersections of the interior common tangents form the triangle PQR.

We see by dragging A B and C that the ratio of the areas of PQR and ABC are independent of the triangle ABC formed by the centers of the circles, but depend only on the radii of the circles.

This document requires an HTML5-compliant browser.
radius A
radius B
radius C
Area PQR
Area ABC

Can you infer the relationship between this ratio and the radii of the circles?

App generated by Geometry Expressions