A One-Line Proof of Leversha’s “Quartet of Isogonal Conjugates” Theorem

Shalosh B. EKHAD and Doron ZEILBERGER

In [L], Gerry Leversha spent quite a few pages, using human ingenuity and lots of previous knowledge, to prove the following elegant theorem (see [W] for the definition of Isogonal Conjugate).

Leversha’s Quartet Theorem ([L]): Let ABCD be a quadrilateral which is not cyclic. Let A^{∗} be the isogonal conjugate of A with respect to ΔBCD, and similarly let B^{∗}, C^{∗}, D^{∗} the isogonal conjugates of ΔADC, ΔABD, ΔABC respectively. Then A,B,C, and D are the circumcenters of ΔB^{∗}C^{∗}D^{∗}, ΔA^{∗}C^{∗}D^{∗}, ΔA^{∗}D^{∗}D^{∗}, and ΔA^{∗}C^{∗}B^{∗}.

Proof: Download the Maple package RENE.txt freely available from Doron Zeilberger’s website

https://sites.math.rutgers.edu/~zeilberg/PG/RENE.txt ,

start Maple, and type read `RENE.txt`: followed by Leversha();, and in one nano-second, you would get that it is true. The Maple code for proving Leversha’s theorem is as follows:

T1 := Te(m, n); T2 := Te(m1, n1); A := [0, 0]; B := T1[3]; C := T2[3];
evalb(DeSq(IsogonalConjugate(m1, n1, B), A) = DeSq(IsogonalConjugate(m, n, C), A));

Explanation

Without loss of generality (by translating, rotating, and shrinking) we can assume that A = (0, 0), D = (1, 0),

tan(\frac{1}{2}^{}B̸AD) = m , tan(\frac{1}{2}^{}A̸DB) = n , tan(\frac{1}{2}^{}C̸AD) = M , tan(\frac{1}{2}^{}A̸DC) = N ,

so now ΔADB and ΔADC are what are called (in RENE.txt), Te(m,n) and Te(M,N). Using the straightforward IsogonalConjugate macro we get B^{∗} and C^{∗} and we just check that the ‘distancesquared’ (DeSq), between A and B^{∗} and A and C^{∗} are the same. Of course, this is also true about the distance between A and B^{∗} and A and D^{∗}, and by the transitivity of the = relation, all three distances are equal and hence indeed A is the circumcenter of ΔB^{∗}C^{∗}D^{∗}.

But using this ‘analytic’ approach (rather than Leversha’s synthetic approach) gives much more. LevershaRadius(m,n,M,N) gives that the exact value of the circumradius of ΔB^{∗}C^{∗}D^{∗} is

\begin{matrix}(N − n) & (Nn + 1) & (m^{2} + 1) & (M^{2} + 1) \\ ((MN − 1)(mn − 1) + (M + N)(n + m)) & (MN(n + m) − mn(M + N) + M + N − m − n) \end{matrix} .

RENE.txt can also tell you the exact locations of B^{∗} and C^{∗}:

B^{∗} = ( \begin{matrix}(Mm + M − m + 1) & (Mm − M + m + 1) & (Nn + 1) & (N − n) \\ (MN(mn − 1) + (M + N)(m + n) − mn + 1) & (MN(m + n) − mn(M + N) + M + N − m − n) \end{matrix} ,

\frac{2 (Nn + 1) (N − n) (Mm + 1) (M − m)}{(MN(mn − 1) + (M + N)(m + n) − mn + 1) (MN(m + n) − mn(M + N) + M + N − m − n)}) ,

C^{∗} = ( \begin{matrix}(Mm + M − m + 1) & (Mm − M + m + 1) & (Nn + 1) & (N − n) \\ (MN(mn − 1) + (M + N)(m + n) − mn + 1) & (MN(m + n) − mn(M + N) + M + N − m − n) \end{matrix} ,

− \frac{2 (Nn + 1) (N − n) (Mm + 1) (M − m)}{(MN(mn − 1) + (M + N)(m + n) − mn + 1) (MN(m + n) − mn(M + N) + M + N − m − n)}) .

References

[L] Gerry Leversha, A quartet of isogonal conjugates, The Mathematical Gazette, 100, issue 548, (July, 2016), 336-341, (Note 100.23). Available from JSTOR.

[W] The Wikipedia Foundation, Isogonal Conjugate

https://en.wikipedia.org/wiki/Isogonal conjugate

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: ShaloshBEkhad at gmail dot com .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .