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aops_forum | Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$ . Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$ . Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC... | 1. **Given**: Triangle $ABC$ with centroid $G$ and circumcenter $O$ such that $GO \perp AG$. Let $A'$ be the second intersection of $AG$ with the circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$, and $E$ the intersection of lines $BA'$ and $AC$.
2. **To Prove**: The circumcenter of t... | [
{
"content": "Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$ . Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$ . Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of ... |
aops_forum | Let $P$ be an interior point of an acute-angled triangle $ABC$ . The line $BP$ meets the line $AC$ at $E$ , and the line $CP$ meets the line $AB$ at $F$ . The lines $AP$ and $EF$ intersect each other at $D$ . Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$ . Show that... | 1. **Setup and Definitions:**
- Let $P$ be an interior point of an acute-angled triangle $ABC$.
- The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$.
- The lines $AP$ and $EF$ intersect each other at $D$.
- Let $K$ be the foot of the perpendicular from the point $D$ to th... | [
{
"content": "Let $P$ be an interior point of an acute-angled triangle $ABC$ . The line $BP$ meets the line $AC$ at $E$ , and the line $CP$ meets the line $AB$ at $F$ . The lines $AP$ and $EF$ intersect each other at $D$ . Let $K$ be the foot of the perpendicular from the point $D$ to the ... |
aops_forum | Let $PQ$ be the diameter of semicircle $H$ . Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$ . Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$ . Prove that $AC$ bisects $\angle PAB$ | 1. **Define the Geometry:**
- Let $PQ$ be the diameter of semicircle $H$.
- Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$.
- Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB \perp PQ$ and is tangent to $O$.
2. **Identify Key Points:**
- Let $X$ be the point of tangency b... | [
{
"content": "Let $PQ$ be the diameter of semicircle $H$ . Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$ . Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\\perp PQ$ and is tangent to $O$ . Prove that $AC$ bisects $\\angle PAB$ ",
"role": "user"
... |
aops_forum | An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed. | 1. **Coloring the Chessboard:**
- We start by coloring the $11 \times 11$ chessboard in a repeating pattern of $1, 2, 3$. The first row will be $12312312312$, the second row will be $23123123123$, and the third row will be $31231231231$. This pattern repeats every three rows.
- The first nine rows will have an eq... | [
{
"content": "An $11\\times 11$ chess board is covered with one $\\boxed{ }$ shaped and forty $\\boxed{ }\\boxed{ }\\boxed{ }$ shaped tiles. Determine the squares where $\\boxed{}$ shaped tile can be placed.",
"role": "user"
},
{
"content": "1. **Coloring the Chessboard:**\n - We start by ... |
aops_forum | In triangle $ABC$ , the midpoints of sides $AB$ and $AC$ are $D$ and $E$ , respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$ . | 1. **Identify Midpoints and Given Conditions:**
Let $D$ and $E$ be the midpoints of sides $AB$ and $AC$ respectively. By definition, $D$ and $E$ divide $AB$ and $AC$ into two equal segments:
\[
AD = DB = \frac{AB}{2} \quad \text{and} \quad AE = EC = \frac{AC}{2}
\]
We are given that the length of side $B... | [
{
"content": "In triangle $ABC$ , the midpoints of sides $AB$ and $AC$ are $D$ and $E$ , respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$ .",
"role": "use... |
aops_forum | Triangle $ ABC$ is inscribed in circle $ \omega$ . The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$ . Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$ . Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$ ) such that $ B_1T \equal{} BT \equal{} C... | 1. **Identify Key Points and Angles:**
- Let $X$ be the intersection of $B_1B$ and $C_1C$.
- We need to show that $\triangle ABC \sim \triangle AB_1C_1$.
2. **Calculate $\angle BXC$:**
- Since $B_1T = BT$ and $C_1T = CT$, the angles $\angle BB_1T$ and $\angle CC_1T$ are equal.
- Therefore, $\angle BXC = 18... | [
{
"content": "Triangle $ ABC$ is inscribed in circle $ \\omega$ . The tangent lines to $ \\omega$ at $ B$ and $ C$ meet at $ T$ . Point $ S$ lies on ray $ BC$ such that $ AS \\perp AT$ . Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$ ) such that $ B_... |
aops_forum | Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$ . Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram. | 1. **Define the angles and use the right-angle property:**
Let $\angle AOB = y$ and $\angle COD = x$. Since $\Delta AMD$ is right-angled at $M$, it follows that:
\[
\frac{x + y}{2} = 90^\circ \implies x + y = 180^\circ
\]
2. **Calculate the distances from the center to the midpoints:**
The midpoints $K$... | [
{
"content": "Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$ . Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.",
"role": "user"
},
{... |
aops_forum | The point $H$ is the orthocentre of a triangle $ABC$ , and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point. | 1. **Identify the orthocenter and altitudes**: Let $H$ be the orthocenter of $\triangle ABC$, and $AD, BE, CF$ be the altitudes of $\triangle ABC$.
2. **Define incenters**: Let $I_1, I_2, I_3$ be the incenters of $\triangle EHF, \triangle FHD, \triangle DHE$ respectively.
3. **Consider external homothetic centers**: ... | [
{
"content": "The point $H$ is the orthocentre of a triangle $ABC$ , and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.",
"role": "user"
},
{
... |
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