Found problems: 325
Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$
[i]Proposed by Alexander Ivanov[/i]
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
Find the radius of a circle inscribed in a triangle with side lengths $4$, $5$, and $6$
Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one
Let $r$ be the radius of the inscribed circle of a right triangle $ABC$. Show that $r$ is less than half of either leg and less than one fourth of the hypotenuse.
Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $
[b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle.
[b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $
[b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $
[i]Cătălin Zârnă[/i]
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.
Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]
A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$
Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
Let $ABC$ be a triangle. Let $K, L$ and $M$ be points on the sides $BC, AC$ and $AB$, respectively, such that $\frac{|AM|}{|MB|}\cdot \frac{|BK|}{|KC|}\cdot \frac{|CL|}{|LA|} = 1$. Prove that it is possible to choose two triangles out of $ALM, BMK, CKL$ whose inradii sum up to at least the inradius of triangle $ABC$.
Let $ABC$ be triangle with the symmedian point $L$ and circumradius $R$. Construct parallelograms $ ADLE$, $BHLK$, $CILJ$ such that $D,H \in AB$, $K, I \in BC$, $J,E \in CA$ Suppose that $DE$, $HK$, $IJ$ pairwise intersect at $X, Y,Z$. Prove that inradius of $XYZ$ is $\frac{R}{2}$ .
Let $AB$ be a triangle and $\Gamma$ the excircle, relative to the vertex $A$, with center $D$. The circle $\Gamma$ is tangent to the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with $BD$ and $CD$, respectively. If $O$ is the point of intersection of $BQ$ and $CP$, show that the distance from $O$ to the line $BC$ is equal to the radius of the inscribed circle in the triangle $ABC$.
Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$. Prove that the triangle is equilateral.
The vertices of a right-angled triangle are on a circle of radius $R$ and the sides of the triangle are tangent to another circle of radius $r$ (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of $R+r$.
In the given figure, $ABCD$ is a square sheet of paper. It is folded along $EF$ such that $A$ goes to a point $A'$ different from $B$ and $C$, on the side $BC$ and $D$ goes to $D'$. The line $A'D'$ cuts $CD$ in $G$. Show that the inradius of the triangle $GCA'$ is the sum of the inradii of the triangles $GD'F$ and $A'BE$.
[asy]
size(5cm);
pair A=(0,0),B=(1,0),C=(1,1),D=(0,1),Ap=(1,0.333),Dp,Ee,F,G;
Ee=extension(A,B,(A+Ap)/2,bisectorpoint(A,Ap));
F=extension(C,D,(A+Ap)/2,bisectorpoint(A,Ap));
Dp=reflect(Ee,F)*D;
G=extension(C,D,Ap,Dp);
D(MP("A",A,W)--MP("E",Ee,S)--MP("B",B,E)--MP("A^{\prime}",Ap,E)--MP("C",C,E)--MP("G",G,NE)--MP("D^{\prime}",Dp,N)--MP("F",F,NNW)--MP("D",D,W)--cycle,black);
draw(Ee--Ap--G--F);
dot(A);dot(B);dot(C);dot(D);dot(Ap);dot(Dp);dot(Ee);dot(F);dot(G);
draw(Ee--F,dashed);
[/asy]
In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.
In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0));
draw(B--M--O--B--C--O--N--C^^N--A--M);
label("$A$", A, dir(90));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$M$", M, dir(90)*dir(B--A));
label("$N$", N, dir(90)*dir(A--C));
label("$O$", O, dir(90));[/asy]
$\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$
The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively.
(Grigory Filippovsky)
On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$.
(M. Volchkevich)
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:
$$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}
\leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$
(Albania)
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.