Problems
Given a triangle $ABC$ with two Brocard points $\Omega_1$ and $\Omega_2$, and the symmedian point $S$. Construct the square $\Omega_1 \Omega_1^{p} \Omega_2 \Omega_2^{p}$. Let $T$ be the harmonic conjugate of $S$ with respect to the pair of points $\Omega_1^{p}$ and $\Omega_2^{p}$. Denote $T^{p}$ and $T^{q}$ as the isogonal conjugate and the isotomic conjugate of $T$, respectively. Prove that the line connecting $T^{p}$ and $T^{q}$ passes through the circumcenter of the triangle $S\Omega_1\Omega_2$.
Happy New Year $2025=(1+2+3+4+5+6+7+8+9)^2$.
LinkLet $ABCD$ be a rectangle, and $P$ an arbitrary point. Denote $P_a, P_b, P_c, P_d$ as the isogonal conjugates of $P$ with respect to the triangles $BCD, CDA, DAB, ABC$, respectively.
1) Then $P_a, P_b, P_c, P_d$ lie on a circle with center $J$. (In fact, this result holds for any cyclic quadrilateral $ABCD$, not just a rectangle.)
2) Let $M$ be the midpoint of $PJ$, and $Q$ the intersection of $P_aP_c$ and $P_bP_d$. Prove that $PQ \parallel MO$, where $O$ is the center of the rectangle $ABCD$.
3) Prove that $OQ$ passes through the Poncelet point of $(P_a, P_b, P_c, P_d)$.
LinkLet $A_1A_2A_3A_4$ be a cyclic quadrilateral inscribed in a circle $\omega$. For $i, j = 1 \ldots 4, \ i < j$, a line $\ell$ intersects the lines $A_iA_j$ at points $P_{ij}$, respectively. Denote $Q_{ij}$ as the perpendicular projections of $P_{ij}$ onto the lines $A_{i+2}A_{j+2}$, where $A_5 = A_1$ and $A_6 = A_2$.
1) Prove that three lines $Q_{12}Q_{34}, Q_{23}Q_{14}, Q_{13}Q_{24}$ are parallel to each other.
2) Prove that the intersections of the pairs of lines $(A_1A_2, A_3A_4),\ (A_2A_3, A_1A_4),\ (A_3A_4, A_1A_2)$ form a triangle that is perspective with the triangle formed by the midpoints of the segments $Q_{12}Q_{34}, Q_{23}Q_{14}, Q_{13}Q_{24}$. Let denote the perspector by $P$.
3) Prove that the line connecting $P$ and Poncelet point of $(A_1,A_2,A_3,A_4)$ is perpendicular to $\ell$.
LinkLet $ABC$ be a triangle with two Brocard points $\Omega_1$ and $\Omega_2$. Let $\Omega_1^{p}$ and $\Omega_2^{p}$ be the isotomic conjugate of $\Omega_1$ and $\Omega_2$ with respect to triangle $ABC$. Let $S$ be the symmedian point of triangle $ABC$. Prove that the centers of three circles $(S\Omega_1\Omega_1^{p})$, $(S\Omega_2\Omega_2^{p})$, and $(ABC)$ are collinear.
LinkIn $\triangle ABC$ with altitudes $AD, BE, CF$, the Euler lines of the triangles $\triangle AEF, \triangle BFD, \triangle CDE$ concur at a point $J$. Let $Fe$ be the Feuerbach point of $\triangle ABC$. Define $\vec{u}, \vec{v}, \vec{w}$ as unit direction vectors of the Euler lines of $\triangle AEF, \triangle BFD, \triangle CDE$, respectively. Prove that it is possible to choose signs $+$ or $-$ such that the vector $\pm\vec{u} \pm\vec{v} \pm\vec{w}$ is perpendicular to $JFe$.
LinkLet $ABCD$ be a convex quadrilateral inscribed in a circle $(O)$. Denote $E, F, G$ as the intersections of the pairs of lines $AB, CD$; $AD, BC$; and $AC, BD$, respectively. Prove that \(\frac{OG \cdot EF}{AC \cdot BD} = \frac{bdk + ack^{-1}}{4S}\) where $AB = a$, $BC = b$, $CD = c$, $DA = d$, and $k = \left| \frac{a^2 - c^2}{b^2 - d^2} \right|$, while $S$ is the area of quadrilateral $ABCD$.
Given a triangle $ABC$ with $I$ as the incenter. A line passing through $A$ intersects the circle $(IBC)$ at points $J$ and $L$. The line $\ell$, passing through $A$ and perpendicular to $BC$, intersects the lines $BJ$, $CJ$, $BL$, and $CL$ at points $P$, $Q$, $R$, and $S$, respectively. Let $AI$ intersect $BC$ at $D$, and let $AK$ be the diameter of $(ABC)$. Prove that the lines $KD$, $\ell$, and the line joining the centers of the circles $(JPQ)$ and $(LRS)$ are concurrent.
