MathLabs

Problem 3

Convex quadrilateral ABCDABCD has ∠ABC=∠CDA=90∘\angle ABC=\angle CDA=90^\circ. Point HH is the foot of the perpendicular from AA to BDBD. Points SS and TT lie on sides ABAB and ADAD, respectively, such that HH lies inside triangle SCTSCT and ∠CHS−∠CSB=90∘\angle CHS-\angle CSB=90^\circ, ∠THC−∠DTC=90∘\angle THC-\angle DTC=90^\circ. Prove that line BDBD is tangent to the circumcircle of triangle TSHTSH.
Step 2 of 6: The circumcentre L of CHT lies on AD
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Similarly, circumcentre L of CHT lies on line AD\text{Similarly, circumcentre } L \text{ of } CHT \text{ lies on line } AD
Detailed analysis

By the same argument applied to the condition ∠THC−∠DTC=90∘\angle THC-\angle DTC=90^\circ, the circumcentre LL of triangle CHTCHT lies on line ADAD.