Related papers: When a triangle is isosceles?
If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle…
We give a trigonometric proof of the Steiner-Lehmus Theorem in hyperbolic geometry. Precisely we show that if two internal bisectors of a triangle on the hyperbolic plane are equal, then the triangle is isosceles.
In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1,…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
In this paper we reformulate Miquel-Steiner's theorem and we obtain Miquel-Steiner's point locus for an arbitrary triangle. We prove that this locus is related to conjugate circles and Brocard's circle. In addition, we obtain…
Let ABC be a triangle with a,b,and c being its three sidelengths. In a 1976 article by Wynne William Wilson in the Mathematical Gazette(see reference[2]), the author showed that angleB is twice angleA, if and only if b^2=a(a+c). We offer…
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we…
Two perpendicular segments which divide a given triangle into 4 regions of equal area is called a quadrisection of the triangle. Leonhard Euler proved in 1779 that every scalene triangle has a quadrisection with its triangular part on the…
A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…
If $P$ is a point inside $\triangle ABC$, then the cevians through $P$ divide $\triangle ABC$ into smaller triangles of various sizes. We give theorems about the relationship between the radii of certain excircles of some of these…
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $\Delta$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a…
If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…
In this article we determine the class of triangles $A_iB_iC_i$ which are orthohomological with a given triangle $ABC$ and inscribed in the triangle $ABC$ (with $A_i \in BC$, $B_i \in CA$ and $C_i \in AB$).
If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.
If $P$ is a point inside $\triangle ABC$, then the cevians through $P$ divide $\triangle ABC$ into six small triangles. We give theorems about the relationships between the radii of the circumcircles of these triangles. We also state some…
In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…
The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are…
For a given triangle $T$ and a real number $\rho$ we define Ceva's triangle $\CT_\rho(T)$ to be the triangle formed by three cevians each joining a vertex of $T$ to the point which divides the opposite side in the ratio $\rho:(1-\rho)$. We…
The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are…
In this article we'll emphasize on two triangles and provide a vectorial proof of the fact that these triangles have the same orthocenter. This proof will further allow us to develop a vectorial proof of the Stevanovic's theorem relative to…