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15 Problems in Geometry

Given a triangle ABC. The centroid G, the orthocenter H, and the center E of the Euler circle belong to the line (the Euler line). Also EH = EO.

Given a quadrilateral $ABCD$. If $G_a$ is the centroid of the triangle $BCD$ opposite the vertex A, and similarly the centroids $G_b$, $G_c$, $G_d$, the centroids of the corresponding triangles, show that the lines $ΑG_a$, $ΒG_b$, $CG_c$ and $DG_d$ intersect at the point $G$.

Given a quadrilateral $ABCD$ inscribed in a circle with center O. If $H_a$ is the orthocenter of $BCD$, $H_b$ is the orthocenter of $ACD$, $H_c$ is the orthocenter of $ABD$, $H_d$ is the orthocenter of $ABC$, show that the lines $AH_a$, $BH_b$, $CH_c$, $DH_d$ intersect at the point H.

Given an inscribed quadrilateral $ABCD$ in a circle with center O. If H is the point in problem 3 and G the point in problem 2, then the points O, H, G are collinear and G is the midpoint of HO. The line HO is called the Euler line of the inscribed quadrilateral. $$\frac{GO}{GH}=-1$$

Given a quadrilateral $ABCD$ inscribed in a circle with center $O$. Let $G_a$, $G_b$, $G_c$, $G_d$ , as in Problem 2. Then, these points are homocyclic and this circle is called the Euler circle for the inscribed quadrilateral $ABCD$. Show that the center $E$ of the Euler circle belongs to the Euler line of $ABCD$.

Let I be the incenter of triangle $ABC$. Then the Euler lines of triangles $AIC,\ AIB,\ BIC$ and $ABC$ converge at a point $S_h$, which is called the Schiffler point¹.

Let $I_a$ be the paracenter of triangle $ABC$. Then the Euler lines of triangles $ABI_a$, $ACI_a$, $BCI_a$ are either concurrent or parallel.

Given an equilateral triangle $ABC$ and a point $P$ in its plane, the Euler lines of the triangles $APB$, $APC$, and $BPC$ are either concurrent or parallel.

Let $ABC$ be a triangle and $P$ a point in the plane. An isogonal² of $CP$ is a line that forms an equal angle with the angle bisector of angle $C$. Show that the isogonals of $AP$, $BP$, and $CP$ intersect at a point $Q$. We will call the point $Q$ the isogonal conjugate of $P$ in triangle $ABC$.

Given a triangle $ABC$ and a point $P$ moving along a line, its isogonal conjugate $Q$ with respect to triangle $ABC$ moves along a conic. The conic is:
- a parabola if it is tangent to the circumcircle,
- an ellipse if the line does not intersect the circumcircle and
- a hyperbola if it intersects the circumcircle.

(Proof oh Problem 10: Greek – English)

Given an equilateral triangle $ABC$ and its excenters $I_a, I_b, I_c$. Let a point $Q$ lie on the locus of the isogonal conjugate $K$ of a point $P$, where $P$ belongs to the circumcircle of triangle $I_a I_b I_c$. Then, the Euler lines of triangles $QAB$, $QAC$, and $QBC$ are parallel.

Let triangle $ABC$ be given and let $P$ be a point in its plane. Prove that the Euler circles (nine-point circles) of the triangles $ABC$, $ABP$, $ACP$, and $BCP$ are concurrent.

Let $ABC$ be a triangle and $P$ a point in its plane. Let $P_a, P_b$ and $P_c$ be the points of intersection of the lines $AP, BP$ and $CP$ with the sides of the triangle, respectively, and let $P’_a, P’_b$ and $P’_c$ be the midpoints of $AP, BP$ and $CP$, respectively. The conic passing through these points also passes through the midpoints of the sides of the triangle. This conic is called the Euler curve of the point $P$.

The Euler curve of a point $P$ passes through the intersection points of the Euler circles of the triangles $ABC$, $ABP$, $ACP$, and $BCP$.

Let $ABC$ be a triangle with orthocenter $H$. Let $O_a$, $O_b$, and $O_c$ denote the midpoints of $AH$, $BH$, and $CH$, respectively.
Let $P$ be an arbitrary point in the plane. The line $PH$ meets the circles with centers $O_a$, $O_b$, and $O_c$ and radii $O_aH$, $O_bA$, and $O_cB$ again at $H_a$, $H_b$, and $H_c$, respectively.
The tangents to these circles at $H_a$, $H_b$, and $H_c$ intersect pairwise at points $A_1$, $B_1$, and $C_1$.
Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.

Lev Emelyanov and Tatiana Emelyanova: A Note on the Schiffler Point, Forum Geometricorum Volume 3 (2003) 113–116.) ↩︎
See wikipedia ↩︎

19.4.2026