How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?
A right triangle has sides with lengths 8, 15 and 17. A circle is inscribed in the triangle, as shown, and we must find the radius of the circle.
Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$.
Prove that triangle $PQR$ is isosceles.
(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.
Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.
As shown below, $ABCD$ is a unit square, $\angle{CBE} = 20^\circ$, and $\angle{FBA} = 25^\circ$. Find the circumstance of $\triangle{DEF}$.
Let $F$ be a point inside $\triangle{ABC}$ such that $\angle{CAF} = \angle{FAB} = \angle{FBC} = \angle{FCA}$, show that the lengths of three sides form a geometric sequence.
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?
Show that the three medians of a triangle intersect at one point.
In $\triangle{ABC}$, let $AD$, $BE$, and $CF$ be the three altitudes as shown. If $AB=6$, $BC=5$, and $EF=3$, what is the length of $BE$?
Let $\triangle{ABC}$ be an acute triangle. If the distance between the vertex $A$ and the orthocenter $H$ is equal to the radius of its circumcircle, find the measurement of $\angle{A}$.
Let $AD$ be the altitude in $\triangle{ABC}$ from the vertex $A$. If $\angle{A}=45^\circ$, $BD=3$, $DC=2$, find the area of $\triangle{ABC}$.
Let $O$ be the centroid of $\triangle{ABC}$. Line $\mathcal{l}$ passes $O$ and intersects $AB$ and $AC$ at $P$ and $Q$, respectively. Point $D$, $E$, and $F$ are on the line $l$ such that $AD\perp l$, $BE \perp l$, and $CF\perp l$. Show that $AD = BE+CF$.
Let $O$ be the incenter of $\triangle{ABC}$. Connect $AO$, $BO$, and $CO$ and extends so that they intersect with $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Let $DE$ intersect $AC$ at $G$, and $DF$ intersects $AB$ at $H$. Show that $G$, $H$ and $O$ are collinear.
Let $ABC$ be an acute triangle. Circle $O$ passes its vertex $B$ and $C$, and intersects $AB$ and $AC$ at $D$ and $E$, respectively. If $O$'s radius equals the radius of $\triangle{ADE}$'s circumcircle, then the circle $O$ must passes $\triangle{ABC}$'s
(A) incenter (B) circumcenter (C) centroid (D) orthocenter.
Let $K$ is an arbitrary point inside $\triangle{ABC}$, and $D$, $E$, and $F$ be the centroids of $\triangle{ABK}$, $\triangle{BCK}$ and $\triangle{CAK}$, respectively. Find the value of $S_{\triangle{ABC}} : S_{\triangle{DEF}}$.
Let $I$ be the incenter of $\triangle{ABC}$. $AI$, $BI$, and $CI$ intersect $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Show that $EF \perp AD$
In $\triangle{ABC}$, $AB=AC$. Extending $CA$ to an arbitrary point $P$. Extending $AB$ to point $Q$ such that $AP=BQ$. Let $O$ be the circumcenter of $\triangle{ABC}$. Show that $A$, $P$, $Q$, and $O$ concyclic.
In triangle ABC, M is median of BC. O is incenter. AH is altitude. MO and AH intersect at E. Prove that AE equal to the radius of incircle
Let $G$ be the centroid of $\triangle{ABC}$, $L$ be a straight line. Prove that $$GG'=\frac{AA'+BB'+CC'}{3}$$ where $A'$, $B'$, $C'$ and $G'$ are the feet of perpendicular lines from $A$, $B$, $C$, and $G$ to $L$.