Some more 3π/4 identities with proof

In a right triangle, all the following identities are true. a, b, c are the sides of the right triangle. r is the inradius of the triangle. m, n are the generating values for primitive Pythagorean triples. #math #maths #mathematics #geometry #trigonometry

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tan^-1(1) + tan^-1(2) + tan^-1(3) = π

The famed 3-4-5 right triangle is used to prove the following trigonometric identity true. #math #maths #mathematics #geometry #trigonometry

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tan^-1 ((a-r)/r) + tan^-1((b-r)/r) = 3π/4

In any right triangle with the incircle constructed, the following trigonometric relationship is true. #math #maths #mathematics #geometry #trigonometry

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An uncommon proof of the Pythagorean Theorem

A proof of the famous theorem using inradius formulae for a right triangle. #math #maths #mathematics #geometry #trigonometry

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r = (a × b)/(a + b + c)

In a right triangle, the inradius can be found by finding the product of the legs and dividing by the perimeter. #math #maths #mathematics #geometry #trigonometry

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n × (m - n) = r

In a right triangle where all the sides are whole numbers, the inradius will be a whole number. Using the formula to generate Pythagorean triple, one can use the generating values to find the length of the inradius. #math #maths #mathematics #geometry #trigonometry

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Proving the existence of the Gergonne point of a triangle

The Gergonne point is where three specific segments are concurrent. These segments come off the vertices and intersect where the incircle is tangent to the side opposite of the vertex. While not as well-known as the most well-known triangle centers, its existence is easily proven with Ceva’s theore

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Proving Ceva’s theorem

Ceva’s theorem is used to prove the concurrency of segments (aka cevians) that come off the vertices and intersect with the opposite side. This proof uses similar triangles to prove the validity of the theorem. #math #maths #mathematics #geometry #trigonometry

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Proving centroid concurrency

The medians of a triangle are concurrent at the centroid. This fact is proven with Ceva’s theorem. #math #maths #mathematics #geometry #trigonometry

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Proving orthocenter concurrency

The orthocenter is where the altitudes of a triangle are concurrent. This fact is proven with Ceva’s theorem. #math #maths #mathematics #geometry #trigonometry

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Proving incenter concurrency

The angle bisectors of any triangle are always concurrent at the incenter. This is the proof of that fact using Ceva’s Theorem. #math #maths #mathematics #geometry #trigonometry

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The law of cotangents

Deriving the law of cotangents using the inradius of a triangle. #math #maths #mathematics #geometry #trigonometry

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cosA × cos B - sin A × sin B = cos(A+B)

This is an obscure derivation of a well-known trigonometric identity. #math #maths #mathematics #geometry #trigonometry

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The triple cotangent identity

Several formulae are derived from locating the incenter of a triangle. The incenter is formed by the angle bisectors meeting concurrently. #math #maths #mathematics #geometry #trigonometry

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The incircle, inradius and equating area formulae

The area of a triangle can be found using the inradius. Using Heron’s formula, one can find the length of the circumradius and the inradius using just the side lengths of the triangle. #math #maths #mathematics #geometry #trigonometry

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Deriving Heron’s formula using the circumradius of the triangle

Heron’s formula is a useful formula to find the area of a triangle using only the triangle’s side lengths. This is a lesser-known derivation of the formula, but quite a cool one nonetheless. #math #maths #mathemati cs #geometry #trigonometry

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Find lengths in terms of the sides

O is the orthocenter of the triangle ABC. The lengths centered around point O and the lengths of the orthic triangle DEF are solved in terms of the side lengths. The area of the triangle is found with Heron’s formula. #math #maths #mathemati cs #geometry #trigonometry

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Side length in the orthic triangle

The orthic triangle is the triangle made by connecting the points where each altitude meets their corresponding side. Since orthocenter is the common meeting point of the quadrilaterals, then three cyclic quadrilaterals are formed, so Ptolemy’s theorem can be used to find the lengths of the sides o

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Simplifying the distance between two points in the heptagonal triangle

In a prior post, I derived the distance formula between the orthocenter and the circumcenter using a complicated argument involving similar triangles. In this post, I used a simple formula to get the required answer. #math #maths #maths #geometry #trigonometry

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The distance between the orthocenter and the circumcenter of a triangle

In any acute triangle, the distance between the orthocenter and the circumcenter is found with quite a nice formula that relate the sides and the circumradius. #math #maths #mathematics #geometry #trigonometry

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