What are circular theorems?
Circle theorems are statements in geometry that state important results related to circles that are used to solve various questions in geometry. Circle theorems in geometry are related to the various components of a circle such as a chord, segments, sector, diameter, tangent, etc. Angles in the same segment of a circle are equal.
Who invented the circle?
Nobody is sure about the inventor of the circle. T he Greeks considered the Egyptians to be the inventors of geometry. Around 650 BC, Thales discover the first theorem related to the circle. In 1700 BC, Rhind Papyrus proposed a method to find the area of a circle.
How do you prove circle theorems?
We can prove circle theorems by using various results in geometry such as the triangle sum theorem, other circle theorems, and theorems based on angles in geometry. Is Circle Theorems Part of Geometry?
How do you find the center of a circle using Thales's theorem?
Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a set square or rectangular sheet of paper larger than the circle. [6] The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2).
How to find tangent line in a circle?
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre ( a, b) and radius r, then the tangent line is perpendicular to the line from ( a, b) to ( x1, y1 ), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at ( x1, y1) determines the value of c, and the result is that the equation of the tangent is
What is tangent in math?
Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point"). All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries. Chord, secant, tangent, radius, and diameter.
What is the ratio of a circle's circumference to its diameter?
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by:
How to prove that a line segment PC will bisect the interior angle APB?
Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB, since the segments are similar:
What is an arc in math?
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle.
How to make a circle with a compass?
The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.
What is the bounding line of a circle?
The bounding line is called its circumference and the point, its centre.
What is the center of a right triangle?
The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse. (Equivalently, a right triangle's hypotenuse is a diameter of its circumcircle.)
Why is the theorem of the sum of angles in a triangle named after Thales?
The theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180°.
What is the circle of a triangle called?
This circle is called the circumcircle of the triangle. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then the triangle is right, and the center of its circumcircle lies on its hypotenuse.
How many vertices are there in a triangle?
For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. ( Sketch of proof. The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle.) This circle is called the circumcircle of the triangle.
How to find the ADC of a right triangle?
Given a right triangle ABC with hypotenuse AC, construct a circle Ω whose diameter is AC. Let O be the center of Ω. Let D be the intersection of Ω and the ray OB . By Thales's theorem, ∠ ADC is right. But then D must equal B. (If D lies inside ∆ ABC, ∠ ADC would be obtuse, and if D lies outside ∆ ABC, ∠ ADC would be acute.)
What is Dante's Paradiso?
Dante's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.
What is the base angle of an isosceles triangle?
Since OA = OB = OC, ∆OBA and ∆OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠OBC = ∠OCB and ∠OBA = ∠OAB.
