MathLabs

Problem 5

Given three distinct points A,B,CA,B,C on a circle KK, construct a point DD on KK such that a circle can be inscribed in the convex quadrilateral ABCDABCD.
Step 4 of 5: Recover the fourth vertex
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D=other tangent from A∩other tangent from CD=\text{other tangent from }A\cap\text{other tangent from }C
Detailed analysis

Draw the circle centered at II tangent to AB,BCAB,BC. From A,CA,C, construct the other tangents to this circle, besides AB,BCAB,BC; their intersection is DD. The locus angle condition makes ABCDABCD cyclic, and the constructed circle is tangent to all four sides, so it is the required incircle.