## Proof Without Words: Nine-Point Circle Property

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.com asked for proof that the center of the nine-point circle […]

## Limits are about the journey …

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Posted 29 April, 2013 by in Classroom, Misc. Math

## 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 […]

Posted 20 April, 2013 by in Classroom, Misc. Math, Open Question

## Proof without Words: Angle Sum and Difference Formulas

One of my favorite images:

Posted 4 March, 2013 by in Classroom, Trigonography

## 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 […]

Posted 16 February, 2013 by in Hedronometry, Open Question