The “Nine-Point Circle” of a triangle \(\triangle ABC\) contains the vertices of the medial triangle (\(\triangle RST \)) and orthic triangle (\(\triangle UVW \)), as well as points \(X\), \(Y\), \(Z\) that bisect segments from \(\triangle ABC\)’s orthocenter (\(H\)) to its vertices. A question on Math.StackExchange.comĀ asked for proof that the center of the nine-point circle […]

## Author Archive

## Proof Without Words: Nine-Point Circle Property

## Limits are about the journey …

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## A Ceva-like Theorem for Tetrahedra

A question at Math.StackExchange asked about “Ceva’s Theorem in three dimensions”. So I derived one. The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with “triple-ratios” of triangle areas, to satisfying effect. Whereas […]

## Proof without Words: Angle Sum and Difference Formulas

One of my favorite images:

## What more I know about hyperbolic tetrahedra

I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF), with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.) The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough […]