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 1 of 5: Determine the incenter's angle condition
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∠BAI+∠BCI=90∘,∠AICreflex=270∘−∠ABC\angle BAI+\angle BCI=90^\circ,\qquad \angle AIC_{\mathrm{reflex}}=270^\circ-\angle ABC
Detailed analysis

If II is the incenter of a cyclic ABCDABCD, then AIAI and CICI bisect the angles at AA and CC. Since opposite angles of a cyclic quadrilateral sum to 180∘180^\circ, this gives ∠BAI+∠BCI=90∘\angle BAI+\angle BCI=90^\circ and hence the reflex ∠AIC=270∘−∠ABC\angle AIC=270^\circ-\angle ABC.