Right triangle

Characterizations

A triangle ${\displaystyle \mathrm{△}ABC}$ with sides ${\displaystyle a\le b<c}$, semiperimeter $s=\frac{1}{2}(a+b+c)$, area ${\displaystyle T,}$ altitude ${\displaystyle h_c}$ opposite the longest side, circumradius ${\displaystyle R,}$ inradius ${\displaystyle r,}$ exradii ${\displaystyle r_a,r_b,r_c}$ tangent to ${\displaystyle a,b,c}$ respectively, and medians ${\displaystyle m_a,m_b,m_c}$ is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.

Sides and semiperimeter

• ${\displaystyle a^2+b^2=c^2(\text{Pythagorean theorem})}$
• ${\displaystyle (s-a)(s-b)=s(s-c)}$
• ${\displaystyle s=2R+r.}$
• ${\displaystyle a^2+b^2+c^2=8R^2.}$

Angles

• ${\displaystyle A}$ and ${\displaystyle B}$ are complementary.
• ${\displaystyle \cos A\cos B\cos C=0.}$
• ${\displaystyle {\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C=2.}$
• ${\displaystyle {\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=1.}$
• ${\displaystyle \sin 2A=\sin 2B=2\sin A\sin B.}$

Area

• ${\displaystyle T=\frac{ab}{2}}$
• ${\displaystyle T=r_a{r}_b=r{r}_c}$
• ${\displaystyle T=r(2R+r)}$
• ${\displaystyle T=\frac{(2s-c{)}^2-c^2}{4}=s(s-c)}$
• ${\displaystyle T=|PA|\cdot |PB|,}$ where ${\displaystyle P}$ is the tangency point of the incircle at the longest side ${\displaystyle AB.}$

Inradius and exradii

• ${\displaystyle r=s-c=(a+b-c)/2}$
• ${\displaystyle r_a=s-b=(a-b+c)/2}$
• ${\displaystyle r_b=s-a=(-a+b+c)/2}$
• ${\displaystyle r_c=s=(a+b+c)/2}$
• ${\displaystyle r_a+r_b+r_c+r=a+b+c}$
• ${\displaystyle r_a^2+r_b^2+r_c^2+r^2=a^2+b^2+c^2}$
• ${\displaystyle r=\frac{r_a{r}_b}{r_c}.}$

Altitude and medians

• ${\displaystyle h_c=\frac{ab}{c}}$
• ${\displaystyle m_a^2+m_b^2+m_c^2=6R^2.}$
• The length of one median is equal to the circumradius.
• The shortest altitude (the one from the vertex with the biggest angle) is the geometric mean of the line segments it divides the opposite (longest) side into. This is the right triangle altitude theorem.

Circumcircle and incircle

• The triangle can be inscribed in a semicircle, with one side coinciding with the entirety of the diameter (Thales' theorem).
• The circumcenter is the midpoint of the longest side.
• The longest side is a diameter of the circumcircle ${\displaystyle (c=2R).}$
• The circumcircle is tangent to the nine-point circle.
• The orthocenter lies on the circumcircle.
• The distance between the incenter and the orthocenter is equal to ${\displaystyle \sqrt{2}r}$.

Trigonometric ratios

The trigonometric functions for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled ${\displaystyle O,}$ ${\displaystyle A,}$ and ${\displaystyle H,}$ respectively, then the trigonometric functions are

${\displaystyle \sin \alpha =\frac{O}{H},\cos \alpha =\frac{A}{H},\tan \alpha =\frac{O}{A},\sec \alpha =\frac{H}{A},\cot \alpha =\frac{A}{O},\csc \alpha =\frac{H}{O}.}$

For the expression of hyperbolic functions as ratio of the sides of a right triangle, see the hyperbolic triangle of a hyperbolic sector.

Special right triangles

The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of ${\displaystyle \frac{1}{6}\pi ,}$ and the isosceles right triangle or 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of ${\displaystyle \frac{1}{4}\pi .}$

Kepler triangle

Let ${\displaystyle H,}$ ${\displaystyle G,}$ and ${\displaystyle A}$ be the harmonic mean, the geometric mean, and the arithmetic mean of two positive numbers ${\displaystyle a}$ and ${\displaystyle b}$ with ${\displaystyle a>b.}$ If a right triangle has legs ${\displaystyle H}$ and ${\displaystyle G}$ and hypotenuse ${\displaystyle A,}$ then

${\displaystyle \frac{A}{H}=\frac{A^2}{G^2}=\frac{G^2}{H^2}=\varphi ,\frac{a}{b}={\varphi }^3,}$

where ${\displaystyle \varphi =\frac{1}{2}(1+\sqrt{5})}$ is the golden ratio. Since the sides of this right triangle are in geometric progression, this is the Kepler triangle.

Thales' theorem

Thales' theorem states that if ${\displaystyle BC}$ is the diameter of a circle and ${\displaystyle A}$ is any other point on the circle, then ${\displaystyle \mathrm{△}ABC}$ is a right triangle with a right angle at ${\displaystyle A.}$ The converse states that the hypotenuse of a right triangle is the diameter of its circumcircle. As a corollary, the circumcircle has its center at the midpoint of the diameter, so the median through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse.

Medians

The following formulas hold for the medians of a right triangle:

${\displaystyle m_a^2+m_b^2=5m_c^2=\frac{5}{4}{c}^2.}$

The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one-half the hypotenuse.

The medians ${\displaystyle m_a}$ and ${\displaystyle m_b}$ from the legs satisfy

${\displaystyle 4c^4+9a^2{b}^2=16m_a^2{m}_b^2.}$

Euler line

In a right triangle, the Euler line contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.

Inequalities

In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times ${\displaystyle (\sqrt{2}-1).}$

In a right triangle with legs ${\displaystyle a,b}$ and hypotenuse ${\displaystyle c,}$

${\displaystyle c\ge \frac{\sqrt{2}}{2}(a+b)}$

with equality only in the isosceles case.

If the altitude from the hypotenuse is denoted ${\displaystyle h_c,}$ then

${\displaystyle h_c\le \frac{\sqrt{2}}{4}(a+b)}$

with equality only in the isosceles case.

Other properties

If segments of lengths ${\displaystyle p}$ and ${\displaystyle q}$ emanating from vertex ${\displaystyle C}$ trisect the hypotenuse into segments of length ${\displaystyle \frac{1}{3}c,}$ then

${\displaystyle p^2+q^2=5{\left(\frac{c}{3}\right)}^2.}$

The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares.

Given any two positive numbers ${\displaystyle h}$ and ${\displaystyle k}$ with ${\displaystyle h>k.}$ Let ${\displaystyle h}$ and ${\displaystyle k}$ be the sides of the two inscribed squares in a right triangle with hypotenuse ${\displaystyle c.}$ Then

${\displaystyle \frac{1}{c^2}+\frac{1}{h^2}=\frac{1}{k^2}.}$

These sides and the incircle radius ${\displaystyle r}$ are related by a similar formula:

${\displaystyle \frac{1}{r}=-\frac{1}{c}+\frac{1}{h}+\frac{1}{k}.}$

The perimeter of a right triangle equals the sum of the radii of the incircle and the three excircles:

${\displaystyle a+b+c=r+r_a+r_b+r_c.}$