Triangle – Die Angst kommt in Wellen
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The right triangle :. The equilateral triangle :. In the equilateral triangle, all the sides are the same length congruent and all the angles are the same size congruent.
Triangle side length rules. Perpendicular bisectors. Circumcenter of a triangle Opens a modal. Circumcenter of a right triangle Opens a modal.
Three points defining a circle Opens a modal. Area circumradius formula proof Opens a modal. Angle bisectors. Incenter and incircles of a triangle Opens a modal.
A triangle is simply defined as a three-sided polygon that consists of three sides also known as edges and vertices. Consider a triangle with its vertices A, B, and C, as shown in the above figure.
A triangle is of two-dimensional shape with its three-sided polygon. It has three sides, and all the sides are made of straight lines.
The common point where two straight lines of a triangle meet are called a vertex. That is why a triangle consists of three vertices.
The circumcircle's radius is called the circumradius. Thales' theorem implies that if the circumcenter is located on a side of the triangle, then the opposite angle is a right one.
If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.
An altitude of a triangle is a straight line through a vertex and perpendicular to i. This opposite side is called the base of the altitude, and the point where the altitude intersects the base or its extension is called the foot of the altitude.
The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H.
The orthocenter lies inside the triangle if and only if the triangle is acute. An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.
The three angle bisectors intersect in a single point, the incenter , usually denoted by I , the center of the triangle's incircle.
The incircle is the circle which lies inside the triangle and touches all three sides. Its radius is called the inradius.
There are three other important circles, the excircles ; they lie outside the triangle and touch one side as well as the extensions of the other two.
The centers of the in- and excircles form an orthocentric system. A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
The three medians intersect in a single point, the triangle's centroid or geometric barycenter, usually denoted by G. The centroid of a rigid triangular object cut out of a thin sheet of uniform density is also its center of mass : the object can be balanced on its centroid in a uniform gravitational field.
The centroid cuts every median in the ratio , i. The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle.
The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle.
It touches the incircle at the Feuerbach point and the three excircles. The orthocenter blue point , center of the nine-point circle red , centroid orange , and circumcenter green all lie on a single line, known as Euler's line red line.
The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.
There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations.
In right triangles , the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides.
The sides of the triangle are known as follows:. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
In our case. This ratio does not depend on the particular right triangle chosen, as long as it contains the angle A , since all those triangles are similar.
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.
Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse. Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse.
Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. However, the arcsin, arccos, etc.
The law of sines , or sine rule, [11] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is.
This ratio is equal to the diameter of the circumscribed circle of the given triangle. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle.
The law of cosines , or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side.
The law of tangents , or tangent rule, can be used to find a side or an angle when two sides and an angle or two angles and a side are known. It states that: [12].
The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy , astronomy , construction , navigation etc.
Calculating the area T of a triangle is an elementary problem encountered often in many different situations.
We recommend that you Topgame to one of the following browsers:. Speak Concede Multiplayer. The Equilateral triangle shown on the Apple Leverkusen has three congruent sides and three congruent angles. Obtuse Triangle.
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