Browse · MathNet
PrintTHE 2002 VIETNAMESE MATHEMATICAL OLYMPIAD
Vietnam 2002 geometry
Problem
In the plane, let be given an isosceles triangle (). A variable circle with center on the line , passes through but does not touch the lines , . Let , be respectively the second points of intersection of the circle with the lines , . Find the locus of the orthocenter of triangle .
Solution
1st case: : the locus is the singleton .
2nd case: : let be the point symmetric to with respect to , be the point symmetric to with respect to then the line is the image of the line under the homothety with center and ratio . Thus the locus of the orthocenter of triangle is where is the image of under the above mentioned homothety; are the points on such that , .
2nd case: : let be the point symmetric to with respect to , be the point symmetric to with respect to then the line is the image of the line under the homothety with center and ratio . Thus the locus of the orthocenter of triangle is where is the image of under the above mentioned homothety; are the points on such that , .
Final answer
If the angle at A is a right angle, the locus of the orthocenter of triangle AMN is the single point A. If the angle at A is not a right angle: let D be the reflection of A across BC. Let d be the image of the line BC under the homothety centered at D with ratio 4·sin^2(A/2). Then the locus of the orthocenter H of triangle AMN is the line d with two points removed: H1 and H2 on d such that DH1 ⟂ DB and DH2 ⟂ DC.
Techniques
Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleHomothetyTriangle trigonometryConstructions and lociTangents