Proof of the Incircle Radius Formula

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[00:00:00]The inscribed circle touches all three sides of a triangle from inside.

[00:00:04]Its radius R equals the area A divided by the semi-perimeter P.

[00:00:09]To prove [music] it, connect the incenter O to each vertex, splitting the triangle into three smaller ones. Each smaller triangle has one side of ABC as its base [music] and the inradius R as its height, since the incircle touches each side at a right angle.

[00:00:24]>> [music] >> So, the total area equals 1/2 AR + 1/2 BR + 1/2 CR.

[00:00:30]Factoring out R gives R times A + B + C over 2. But, A + B + C over 2 is the semi-perimeter P. So, A equals R times P, and dividing both sides by P gives R equals A over P. The formula [music] is proved, and it holds for any triangle.