# Mind Map: Theorems

Theorem
In mathematics, a theorem is a statement, often stated in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions.

The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.

There are many important theorems in Euclidean geometry, which is the study of geometry based on the assumptions of Euclid's Elements. Some of the most well-known and important Euclidean theorems include:

Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Euclid's parallel postulate: Given a line and a point not on the line, there exists exactly one line through the point that is parallel to the given line.

The law of cosines: In a triangle, the square of the length of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.

The law of sines: In a triangle, the ratio of the length of a side to the sine of the opposite angle is the same for all three sides.

The angle bisector theorem: In a triangle, the angle bisector of an angle divides the opposite side into two segments that are proportional to the lengths of the other two sides.

The theorem of the intersecting chords: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

The theorem of the inscribed angles: The measure of an angle formed by two chords that intersect at a point on the circumference of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

The theorem of the central angles: The measure of an angle formed by two radii of a circle is equal to the measure of the arc intercepted by the angle.

These theorems are just a small selection of the many important theorems in Euclidean geometry. They cover a wide range of topics, including triangles, circles, and angles, and provide insights into the relationships between various geometric figures and measurements.

Mind Map
The mind map above is an image-centered diagram that represents connections between various topics and concepts related to math theorems.

The visual mind map can also include images or diagrams to help illustrate the concepts and theorems being discussed. This can help students to better understand the connections between different ideas in Euclidean geometry and to visualize how the various theorems and concepts relate to each other.