Question

Let P be a point on the circumcircle of triangle ABC and let PL, PM and PN be perpendiculars on BCAC and BA produced respectively. Prove that the points L, M and N are collinear.​

Answer 1

To prove that the points L, M and N are collinear

1: Assume a right angle triangle ABC with angle B=90°, AB=BC and D as midpoint of AC.

2: Circumcircle of this triangle will be a circle with centre D and diameter AC.

3: Let P be a point on this circle such that chord PB forms a diameter of the circle.

4: Let a perpendicular from P on AC be PM, which implies that point M= D and will lie on side AC.

5: Let a perpendicular from P on BC be PL, this point L will apparently lie on point C i.e C=L.

6: Similarly a perpendicular from P on BA be PN, this point N will apparently lie on point A i.e A=N.

6: From steps 4,5 and 6 it is observed that points L, M and N lie on side AC.

7: Hence points L, M and N are collinear.