Found problems: 333
In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$
On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$.
The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$.
Prove that $BU = BA$ if, and only if, $CP = CA$.
(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$.
The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$.
Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$.
Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.
In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled.
Liana Topan
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.
Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.
Let $ABC$ have side lengths $3$, $4$, and $5$. Let $P$ be a point inside $ABC$. What is the minimum sum of lengths of the altitudes from $P$ to the side lengths of $ABC$?
Let $ABC$ be a right angled triangle with $\angle A = 90^o$ and $AD$ its altitude. We draw parallel lines from $D$ to the vertical sides of the triangle and we call $E, Z$ their points of intersection with $AB$ and $AC$ respectively. The parallel line from $C$ to $EZ$ intersects the line $AB$ at the point $N$. Let $A' $ be the symmetric of $A$ with respect to the line $EZ$ and $I, K$ the projections of $A'$ onto $AB$ and $AC$ respectively. If $T$ is the point of intersection of the lines $IK$ and $DE$, prove that $\angle NA'T = \angle ADT$.
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Construct a right triangle, given two medians drawn to its legs.
Suppose triangle $ABC$ is right-angled at $B$. The altitude from $B$ intersects the side $AC$ at point $D$. If points $E$ and $F$ are the midpoints of $BD$ and $CD$, prove that $AE \perp BF$.
An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.
Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, find the measure of $\angle MBC$.
Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$.
(Walther Janous)
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.
Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$.
Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.
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[i]Edited by Orl.[/i]
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In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that:
a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$.
b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.
Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than $\sqrt2$?