Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one.

image:
www.aanda.org

More is true: 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.

image:
slideplayer.com

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 B.C.E.

image:
s-media-cache-ak0.pinimg.com

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e.

image:
study.com

An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to.

image:
etc.usf.edu

The orthocenter lies inside the triangle if and only if the triangle is acute.

image:
jwilson.coe.uga.edu

A triangle is one of the basic shapes of geometry: a polygon with three corners (or vertices) and three sides (or edges) that are straight line segments.

image:
www.math-salamanders.com

Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other.

image:
slideplayer.com

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.

image:
vt-s3-files.s3.amazonaws.com

The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices.

image:
dj1hlxw0wr920.cloudfront.net

What follows is a selection of frequently used formulae for the area of a triangle.

The altitude of a triangle can be found through an application of trigonometry.

image:
upload.wikimedia.org

where s = Ѕ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.

image:
www.gogeometry.com

The three altitudes intersect in a single point, called the orthocenter of the triangle.

image:
jwilson.coe.uga.edu

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined.

image:
1.bp.blogspot.com

Calculating the area of a triangle is an elementary problem encountered often in many different situations.

image:
mrkwon.weebly.com

The incircle is the circle which lies inside the triangle and touches all three sides.

image:
mathworld.wolfram.com

where is area, is the length of the base of the triangle, and is the height or altitude of the triangle.

image:
usercontent2.hubstatic.com

A non-planar triangle is a triangle which is not contained in a (flat) plane.

image:
cgm.cs.mcgill.ca

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side.

image:
slideplayer.com

Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

image:
slideplayer.com

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.

image:
www.varsitytutors.com

Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion.

image:
www.cut-the-knot.org

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.

image:
bam.files.bbci.co.uk

The three angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle.

image:
slideplayer.com

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it (i.e.

image:
1.bp.blogspot.com

The three symmedians intersect in a single point, the symmedian point of the triangle.

image:
jwilson.coe.uga.edu

The shape of the triangle is determined by the lengths of the sides alone.

image:
www.mathamazement.com

image:
mrkwon.weebly.com

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.

image:
jwilson.coe.uga.edu

image:
www.math-salamanders.com

Triangles = Strength. Triangles are the strongest shape! Any weight placed on them is evenly distributed between all three sides. A triangle in any fashion can always represent strength. Place a triangle symbol on your altar or wear one as a reminder of how strong you are.Aug 21, 2017

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. If that is not true, then it is not possible to construct a triangle with the given side lengths.

The properties of equilateral triangles: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 180 degrees (from above)... And there are three angles...

A triangle with all sides equal is called equilateral, a triangle with two sides equal is called isosceles, and a triangle with all sides a different length is called scalene.

A triangle is scalene if all of its three sides are different (in which case, the three angles are also different). If two of its sides are equal, a triangle is called isosceles. A triangle with all three equal sides is called equilateral.

A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal.

When you have thought about all four click on "check your answers" and see if you were right. An equilateral triangle has got 3 sides of equal length and 3 angles that are equal. An Isosceles triangle has got two sides of equal length and 2 angles equal.