Found problems: 509
Triangle $ABC$ is inscribed in circle $(O)$. $ P$ is a point on arc $BC$ that does not contain $ A$ such that $AP$ is the symmedian of triangle $ABC$. $E ,F$ are symmetric of $P$ wrt $CA, AB$ respectively . $K$ is symmetric of $A$ wrt $EF$. $L$ is the projection of $K$ on the line passing through $A$ and parallel to $BC$. Prove that $PA=PL$.
Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.
In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.
A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the intersection point of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.
Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?
In the triangle $ABC$ it is known that $\angle ACB> 90 {} ^ \circ$, $\angle CBA> 45 {} ^ \circ$. On the sides $AC$ and $AB$, respectively, there are points $P$ and $T$ such that $ABC$ and $PT = BC$. The points ${{P} _ {1}}$ and ${{T} _ {1}}$ on the sides $AC$ and $AB$ are such that $AP = C {{P} _ {1}}$ and $AT = B {{T} _ {1}}$. Prove that $\angle CBA- \angle {{P} _ {1}} {{T} _ {1}} A = 45 {} ^ \circ$.
(Anton Trygub)
Let $C_1,C_2,..., C_n$ , $(n\ge 3)$ be circles having a common point $M$. Three straight lines passing through $M$ intersect again the circles in $A_1, A_2,..., A_n$ ; $B_1,B_2,..., B_n$ and $X_1,X_2,..., X_n$ respectively. Prove that if
$$A_1A_2 =A_2A_3 =...=A_{n-1}A_n$$ and $$B_1B_2 =B_2B_3 =...=B_{n-1}B_n$$ then $$X_1X_2 =X_2X_3 =...=X_{n-1}X_n.$$
In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.
Consider $\vartriangle ABC$, with $AD$ bisecting $\angle BAC$, $D$ on segment $BC$. Let $E$ be a point on $BC$, such that $BD = EC$. Through $E$ we draw the line $\ell$ parallel to $AD$ and consider a point $ P$ on it and inside the $\vartriangle ABC$. Let $G$ be the point where line $BP$ cuts side $AC$ and let F be the point where line $CP$ to side $AB$. Show that $BF = CG$.
Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $\angle BAC$, if $WO = WH$.
(O. Clurman)
On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.
The points $M$ and $N$ are located respectively on the diagonal $(AC)$ and the side $(BC)$ of the square $ABCD$ such that $MN = MD$. Determine the measure of the angle $MDN$.
On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.
Given an acute-angled $\vartriangle AB$C with altitude $AH$ ( $\angle BAC > 45^o > \angle AB$C). The perpendicular bisector of $AB$ intersects $BC$ at point $D$. Let $K$ be the midpoint of $BF$, where $F$ is the foot of the perpendicular from $C$ on $AD$. Point $H'$ is the symmetric to $H$ wrt $K$. Point $P$ lies on the line $AD$, such that $H'P \perp AB$. Prove that $AK = KP$.
In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
[img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]
Let $M$ be the point of intersection of the diagonals, and $K$ be the point of intersection of the bisectors of the angles $B$ and $C$ of the convex quadrilateral $ABCD$. Prove that points $A, B, M, K$ lie on the same circle if the following relation holds: $|AB|=|BC|=|CD|$
Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$.
Note: The circumcircle of a triangle is the circle that passes through its three vertices.
A point $ N $ is marked on the median $ CM $ of the triangle $ ABC $ so that $ MN \cdot MC = AB ^ 2/4 $. Lines $ AN $ and $ BN $ intersect the circumcircle $ \triangle ABC $ for the second time at points $ P $ and $ Q $, respectively. $ R $ is the point of segment $ PQ $, nearest to $ Q $, such that $ \angle NRC = \angle BNC $. $ S $ is the point of the segment $ PQ $ closest to $ P $ such that $ \angle NSC = \angle ANC $. Prove that $ RN = SN $.
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.
Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.
Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.
We are given an acute-angled triangle $ABC$ with $AB < AC < BC$. Points $K$ and $L$ are chosen on segments $AC$ and $BC$, respectively, so that $AB = CK = CL$. Perpendicular bisectors of segments $AK$ and $BL$ intersect the line $AB$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at point $M$. Prove that $AK + KM = BL + LM$.
Poland
Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.