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Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

5 votes
1 answer
526 views

Twin circles in a quadrilateral

The circumcenters of the four triangles of a complete quadrilateral along with the two points of completion form two congruent circles (in black). Surely this must've been done before - what's the fo …
1 vote
0 answers
158 views

A certain circle formed by perpendiculars

If six points are chosen, two points on each side of a triangle, such that they have the same ratio of distances to vertices, then the perpendicular lines through those points meet at six concyclic p …
0 votes
0 answers
77 views

Coordinates of the centers of the insphere and circumsphere

Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one deter …
0 votes
2 answers
172 views

Radical line of two ellipses

The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation $$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x …