This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 13

1975 Chisinau City MO, 90
Tags: right triangle , median , triangle construction , geometry
Construct a right-angled triangle along its two medians, starting from the acute angles.

1991 Tournament Of Towns, (292) 2
Tags: triangle construction , geometric construction , geometry
Two points $K$ and $L$ are given on a circle. Construct a triangle $ABC$ so that its vertex $C$ and the intersection points of its medians $AK$ and $BL$ both lie on the circle, $K$ and $L$ being the midpoints of its sides $BC$ and $AC$.

1941 Moscow Mathematical Olympiad, 086
Tags: geometry , triangle construction , geometric construction
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

1938 Moscow Mathematical Olympiad, 041
Tags: geometry , geometric construction , triangle construction
Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

1962 All Russian Mathematical Olympiad, 018
Tags: geometric construction , triangle construction , geometry
Given two sides of the triangle. Construct that triangle, if medians to those sides are orthogonal.

1949-56 Chisinau City MO, 33
Tags: triangle construction , geometry
Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.

1936 Moscow Mathematical Olympiad, 023
Tags: rectangle , triangle construction , geometric construction , isosceles
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.

1941 Moscow Mathematical Olympiad, 090
Tags: right triangle , geometric construction , triangle construction , geometry
Construct a right triangle, given two medians drawn to its legs.

1941 Moscow Mathematical Olympiad, 078
Tags: triangle construction , geometric construction , geometry
Given points $M$ and $N$, the bases of heights $AM$ and $BN$ of $\vartriangle ABC$ and the line to which the side $AB$ belongs. Construct $\vartriangle ABC$.

1974 Chisinau City MO, 71
Tags: geometry , centroid , triangle construction
The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

1940 Moscow Mathematical Olympiad, 068
Tags: triangle construction , geometric construction , circumcenter , geometry
The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

1935 Moscow Mathematical Olympiad, 013
Tags: geometry , geometric construction , triangle construction
The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.

1941 Moscow Mathematical Olympiad, 071
Tags: geometry , triangle construction
Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.