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 2 of 5: Construct the circle through A and C

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\angle AXC=90^\circ+\angle ABC

Detailed analysis

Construct the circle $\Gamma$ through $A,C$ for which points $X$ on the selected arc satisfy $\angle AXC=90^\circ+\angle ABC$ . Take a diameter $AE$ of $K$ : the angle relation gives $\angle CAE=90^\circ-\angle ABC$ , so $AE$ is tangent to $\Gamma$ at $A$ . The center of $\Gamma$ is the intersection of the perpendicular to $AE$ through $A$ and the perpendicular bisector of $AC$ .