Existence of the triangle with sides $\sqrt{bc}$, $\sqrt{ca}$, $\sqrt{ab}$.

Given the sides $a$,$b$,$c$ of a rectangle we would like to know if a triangle exists whose sides are $\sqrt{bc}$, $\sqrt{ca}$, $\sqrt{ab}$, the geometric means of $a$, $b$, $c$. 

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Existencia del triángulo con lados $\sqrt{bc}$, $\sqrt{ca}$, $\sqrt{ab}$

Construction of X5000 and X5001

We give a construction of points X5000 and X5001.

Construction of X5000 and X5001

The crosspoint

We study some loci related to the construction known as crosspoint of two given points. The main result is as follows:

Let $ABC$ a triangle, $P$ a point and $L$ the trilinear polar of some point $M$. Then the locus of points $Q$ for which the crosspoint of $P$ and $Q$ lies on $L$ is the conic with perspector the crosspoint of  $P$ and $M$.

See details at The crosspoint

A property of X7090 and X14121

Given a triangle ABC, their points X7090 and X14121 are the complement of X176 and X175 respectively.

They are the centers of the circles tangent internally to the three circles with BC, CA, AB as diameters.

See details at A property of  X7090 and X14121