Question

Let ABC be a triangle and let H be its orthocentre. Let D be the point different from C in which the line CH intersects the circumcircle of ABC .
Prove that the triangle AHD is isosceles.

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Answer 1

Answer:

angle chasing problem

Step-by-step explanation:

1) We make line CH intersects the circumcircle of ABC at point D. Since angles ABC and ADC share a same segment AC in the circumcircle, we can get angles ABC=ADC

2) make line AH intersects BC at point E, and CH intersects AB at point F. Since H is the orthocentre, CF is perpendicular to AB , and AE is perpendicular to CB.

3) in triangle CFB, we can get angles FCB+CBF=90 degrees, in triangle CHE, we can get angles CHE+HCE=90 degrees., so we know angle CHE=CBA.

4) Since angle AHD=CHE, we can know angle AHE=CHE=CBA=CDA.

5) prove