Problem 5

Given three distinct points

A,B,C

on a circle

K

, construct a point

D

K

such that a circle can be inscribed in the convex quadrilateral

ABCD

Step 3 of 5: Locate the incenter

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I=\Gamma\cap\operatorname{bisector}_{\mathrm{int}}(\angle ABC)

Detailed analysis

Choose the intersection $I$ of $\Gamma$ with the internal angle bisector of $\angle ABC$ that lies in the intended convex configuration. The locus condition gives the required angle at $I$ , while the angle-bisector condition makes the distances from $I$ to lines $AB$ and $BC$ equal.