MathDB

14

Part of 2003 National High School Mathematics League

Problems(1)

Complex Number

Source: 2003 National High School Mathematics League, Exam One, Problem 14

3/16/2020
A,B,CA,B,C are points that three complex numbers z0=ai,z1=12+bi,z2=1+ci(a,b,c∈R)z_0=a\text{i},z_1=\frac{1}{2}+b\text{i},z_2=1+c\text{i}(a,b,c\in\mathbb{R}) refer to on complex plane (not collinear). Prove that curve Z=Z0cos⁡4t+2Z1cos⁡2tsin⁡2t+Z2sin⁡4t(t∈R)Z=Z_0\cos^4t+2Z_1\cos^2t\sin^2t+Z_2\sin^4t(t\in\mathbb{R}) has only one common point with the perpendicular bisector of ACAC, and find the point.
complex numbersgeometryperpendicular bisector