Radii of In-circle and Ex-circles of a Right- Angled Triangle

In this article, I provide a relation connecting the lengths of the tangents from the vertices of a right-angled triangle to its incircle and ex-circles, in terms of its inradius and ex-radii. I give a geometric proof as well as an analytic proof.

A standard result which will be used repeatedly is the following: Given a circle and a point outside it, the lengths of the two tangents that can be drawn from the point to the circle have equal length. A list of more such results and formulas of relevance is provided at the end of the article.

The following nomenclature should be noted. Other than the incircle of a triangle, three other circles can be drawn that touch the sidelines of a triangle. These are called the ex-circles of the triangle. The ex-circle opposite vertex A is known as the ‘A ex-circle’, and likewise for the two other ex-circles. The radius r a of the A ex-circle is called the ‘A ex-radius’, and similarly for the radii of the two other ex-circles.

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