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Today we shall have a (very brief) look at the $latex D-$excircle of the orthic triangle which appeared recently on the Vietnam Team Selection Test. This configuration can be transposed to the standard excircle configuration by shifting the reference triangle to the orthic triangle, but certain results are interpreted more naturally in an orthic setting.… Continue reading D-Excircle of the Orthic triangle
Today we shall explore a more classical configuration which has been studied in various settings -- most notably in terms of inversion. We consider $latex \omega_1$,$latex \omega_2$,$latex \omega_3$ and $latex \omega_4$ (with centers $latex O_1$, $latex O_2$, $latex O_3$ and $latex O_4$) to be four circles such that $latex \omega_i$ and $latex \omega_{i+1}$ are externally… Continue reading Four Kissing Circles
Today we shall have a look at another orthocentric configuration, that of the Miquel point of the quadrilateral formed by the intersections of the tangent to circle $latex (BHC)$ at $latex H$ with the sides of $latex \triangle ABC$. Many thanks to Yangqin Yang for contributing several proofs and for saving me from crashing out.… Continue reading Miquel point of a quadrilateral formed by the H-tangent to (BHC)
Today we shall have a look at a generalized configuration, with certain noteworthy special cases. Problems from this configuration seem to not yet have appeared in olympiads, with the only problem I am aware of being due to Ethan Wang. Also, shoutouts to Yanqing Yang for several key ideas and claims. Since most points are… Continue reading Intersection of a circle centered on a cevian
Today we shall look at something different from usual, and turn our attention to a famous triangle center, named after Benjamin Bevan -- the circumcenter of the excentral triangle or the Bevan point $latex Be$, which is Kimberling triangle center $latex X_{40}$. A reasonable amount of work (mainly analytical) has been done regarding this point… Continue reading The Bevan Point
Today we shall look at a rather interesting configuration which has appeared in only a handful of problems to date with rather surprising and occasionally obscure results. Our work shall explore the tangents from the point $latex T$ which is defined as the intersection of the tangents from $latex B$ and $latex C$ to the… Continue reading Tangents to the Incircle
Today we shall have a look at a simple configuration which is moderately prevalent in modern Olympiads but has not received much attention. Our work shall involve the feet of the perpendiculars from the foot of the $latex A-$altitude to the sides $latex AB$ and $latex AC$. As usual, notation is borrowed from the diagram… Continue reading Feet to the sides from the A-foot
In this article, we shall explore some properties of the intersections two highly celebrated circles in the world of Olympiad geometry. Although the individual circles are thoroughly explored and considered mostly 'well-known', the configuration involving their intersections is still relatively unexplored, with only two notable examples appearing on an Olympiad to date, as far as… Continue reading Intersections of the Nine-Point Circle and (BHC)
Today we will revisit a rather old but underrated configuration which has certain beautiful properties. Our work shall revolve around the points $latex E$ on $latex AC$ and $latex F$ on $latex AB$ such that $latex AE=AF$ and points $latex E,H$ and $latex F$ lie on a straight line.Shoutouts to raosicheng for suggesting an overwhleming… Continue reading Isosceles triangle with the orthocenter on the base
Today we will explore the properties of a certain well-known point with respect to the intouch triangle. The Why Point, originally seen in the famous 2011 G4, is a thoroughly explored configuration with a plethora of interesting properties. Among the many perspectives of the Why Point configuration is the Why Point of the Intouch triangle.Here… Continue reading Why Point of the Intouch Triangle
Angular Momentum 