Tag Archives: triangle

Spectral Golden Spiral II

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Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles

There are two isosceles triangles which are related to the golden ratio, [1 + sqrt(5)]/2, and I used to refer to them as the “golden acute isosceles triangle” and the “golden obtuse isosceles triangle,” before I found out these triangles … Continue reading

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On Triangle Congruence, and Why SSA Does Not Work

Those who have taught geometry, when teaching triangle congruence, go through a familiar pattern. SSS (side-side-side) triangle congruence is usually taught first, as a postulate, or axiom — a statement so obvious that it requires no proof (although demonstrations certainly … Continue reading

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A Polyhedral Demonstration of the Fact That Twenty Times Four Is Eighty

The Platonic solid known as the icosahedron has twenty triangular faces. This polyhedron resembles the icosahedron, but with each of the icosahedron’s triangles replaced by a panel of four faces:  three isosceles trapezoids surrounding a central triangle. Since (20)(4) = … Continue reading

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The 9-81-90 Triangle

In a previous post (right here), I explained the 18-72-90 triangle, derived from the regular pentagon. It looks like this: I’m now going to attempt derivation of another “extra-special right triangle” by applying half-angle trigonometric identities to the 18º angle. After … Continue reading

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Three Superimposed Hexagonal Tessellations Can Make a Triangular Tessellation

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A Comparison of the Areas of Some of the Triangles Formed By Connecting Three Noncollinear Triangle Centers

The five most well-known triangle centers are the centroid (where a triangle’s medians meet), the orthocenter (where the lines containing the altitudes meet), the incenter (where a triangle’s three interior angle bisectors meet), the circumcenter (where the perpendicular bisectors of … Continue reading

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