Question

Consider ∆ABC whose circumcircle is |z| = r in the argand plane with A, B, C be represented by complex numbers a, b, c respectively. The foot of altitudes from A, B, C meet the opposite side at D, E and F and the altitudes when produced meet the circle |z| = r at L, M and N respectively then which of following is/are correct?

• A. centroid of ∆LMN is $\frac{-(a^2b^2 + b^2c^2 + c^2a^2)}{3abc}$
• B. the mirror image of orthocenter of ∆ABC with respect to BC lies on |z| = r
• C. the mirror image of orthocenter of ∆ABC with respect to BC is $\frac{-bc}{a}$
• D. none of these

Answer 1

Answer: (A), (B), (C)

Options (A), (B), and (C) are correct.

Answer 2

It is known that if the vertices of ∆ABC lie on the circle |z| = r then the following facts hold:

1. The points where the altitudes (extended) meet the circle are the reflections of the orthocenter in the sides. In particular, the reflection of the orthocenter H across BC lies on the circle |z| = r. This proves option (B).

2. In complex number geometry (with |a| = |b| = |c| = r), one can show that the reflection of H in side BC is given by L = –(bc)/a. Analogously, one finds M = –(ca)/b and N = –(ab)/c. This verifies option (C).

3. The centroid of ∆LMN is (L + M + N)/3 = [–(bc)/a – (ca)/b – (ab)/c] / 3. Multiplying numerator and denominator by abc, we get = –(a²b² + b²c² + c²a²)/(3abc), which is exactly option (A).