A triangle in geometric progression

Draw a segment perpendicular to a side at the vertex (in orange) and let it intersect with the side above the vertex of the regular heptagon. Connect the top of the perpendicular segment to the vertices that are further away from where the perpendicular segment is and don’t share a side with the or

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1-√2-2 isosceles trapezoid in the regular heptagon

The first pic shows how the trapezoid is constructed and the reasoning on why it exists. The other pictures show the segments needed to construct the trapezoid. Red segments have a length of 1, blue segments have a length of √2 and green segments have a length of 2. Furthermore, the orange segments

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2 in the regular heptagon, part 3

Another way to derive the value of the segment labeled m (the segment between the points where ωa meets the side of the heptagon) is with the Pythagorean theorem. The segments labeled m, ωan and c-b make a right triangle and from there we can get the value of 2. #math #maths #mathematics

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2 in the regular heptagon, part 2

Since the regular heptagon is symmetric around the top vertex, two reflected heptagonal triangles can be formed near the top of the shape. Thus, from where the angle bisectors from the smallest angles in each triangle to where they meet the opposite side lengths are reflected points across the line

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2 in the regular heptagon, part 1

The distance from where the angle bisector coming off the smallest angle in the heptagonal triangle intersects with the shortest side (which is also a side length of the regular heptagon) to the opposite bottom of the vertex of the heptagon is exactly 2. Not many people know that fact. #math

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√2 in the regular heptagon, part 3

Where the two diagonals cross in the heptagonal triangle is the incenter of the triangle. The distance from the incenter to the closest bottom vertex is equal to the √2. #math #maths #mathematics #geometry #trigonometry

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√2 in the regular heptagon, part 2

In the heptagonal triangle, ωa is the angle bisector coming off the smallest angle and it intersects the side length of the regular heptagon, which is also the smallest side of the triangle. The distance from the bottom vertex to the intersection of the angle bisector and the side is equal to √2. T

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√2 in the regular heptagon, part 1

Not many people know that √2 is a component of the regular heptagon. By using established identities and the law of cosines, the √2 shows itself here as the distance between the intersection of the long diagonals and the vertex below. #math #maths #mathematics #geometry #trigonomet

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