state and prove Thales theorm
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IN geometry thales theorem state that if A,B and C are the distinct points on circle where the line AC is a diameter, then the angle ABC iis a triangle
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Using the diameter of a circle as the base of a triangle with the apex on that circle, means that this triangle will be a right triangle and the diameter will be the hypotenuse of that triangle.
“Triangle in a Circle”
“The diameter of a circle subtends a right angle to any point on the circle”. (The word “subtends” means that it “creates an angle at a distant point”).
The diagram shows it more clearly. Position ‘O’ is the center of the circle. Line AB is the diameter of the circle. Point ‘c’ is on the circumference.
Thales’ Theorem says that the angle ACB of the triangle ACB is a right angle…regardless of where ‘C’ may be placed on the circumference.
To Prove Thales’ Theorem
To prove Thales’ Theorem, let’s start by adding another line, CO, to the diagram.
Thales Triangle with Notes: Image by Mike DeHaan (Click for a closer look)
Preparing for the Proof of Thales’ Theorem
The point ‘O’ is the center of a circle with radius of length ‘r’. AB is a diameter with ‘O’ at the center, so length(AO) = length(OB) = r. Point C is the third point on the circumference. We have triangles OCA and OCB, and length(OC) = r also.
Triangle OCA is isosceles since length(AO) = length(CO) = r. Therefore angle(OAC) = angle(OCA); let’s call it ‘α‘ (“alpha”). Likewise, triangle OCB is isosceles since length(BO) = length(CO) = r. Therefore angle(OBC) = angle(OCB); let’s call it ‘β‘ (“beta”).
(If anyone asks, we can get into a discussion of this property of isosceles triangles at another time. That’s the purpose of the “comments” section, in my humble opinion.)
Since AOB is a straight line, angle (AOB) = 180. So angle(AOC) + angle(COB) = 180. Let’s record this fact as angle(COB) = 180 – angle(AOC).
The sum of the interior angles of a triangle is 180. So in triangle OCA, the sum of angles OAC + OCA + AOC = 180. As we said before, this is α+α + angle(AOC) = 180, or 2*α + angle(AOC) = 180.
Likewise, in triangle OCB, angle(OCB) + angle(OBC) + angle(COB) = 180. As noted, we can say β+β + angle(COB) = 180, or 2*β + angle(COB) = 180.
First Proof of Thales’ Theorem
We also see that triangle ACB is a triangle, so its interior angles add up to 180 also. So angle(BAC) + angle(ABC) + angle(ACB) = 180. That was α+β + angle(ACB) = 180.
But clearly angle(ACB) is the sum of angle(ACO) + angle(OCB), which are values α and β. So α+β + α+β = 180. So 2*(α+β) = 180. Dividing both sides by 2 shows that α+β = 90. This is precisely what Thales’ Theorem says, so it is proven.
“Triangle in a Circle”
“The diameter of a circle subtends a right angle to any point on the circle”. (The word “subtends” means that it “creates an angle at a distant point”).
The diagram shows it more clearly. Position ‘O’ is the center of the circle. Line AB is the diameter of the circle. Point ‘c’ is on the circumference.
Thales’ Theorem says that the angle ACB of the triangle ACB is a right angle…regardless of where ‘C’ may be placed on the circumference.
To Prove Thales’ Theorem
To prove Thales’ Theorem, let’s start by adding another line, CO, to the diagram.
Thales Triangle with Notes: Image by Mike DeHaan (Click for a closer look)
Preparing for the Proof of Thales’ Theorem
The point ‘O’ is the center of a circle with radius of length ‘r’. AB is a diameter with ‘O’ at the center, so length(AO) = length(OB) = r. Point C is the third point on the circumference. We have triangles OCA and OCB, and length(OC) = r also.
Triangle OCA is isosceles since length(AO) = length(CO) = r. Therefore angle(OAC) = angle(OCA); let’s call it ‘α‘ (“alpha”). Likewise, triangle OCB is isosceles since length(BO) = length(CO) = r. Therefore angle(OBC) = angle(OCB); let’s call it ‘β‘ (“beta”).
(If anyone asks, we can get into a discussion of this property of isosceles triangles at another time. That’s the purpose of the “comments” section, in my humble opinion.)
Since AOB is a straight line, angle (AOB) = 180. So angle(AOC) + angle(COB) = 180. Let’s record this fact as angle(COB) = 180 – angle(AOC).
The sum of the interior angles of a triangle is 180. So in triangle OCA, the sum of angles OAC + OCA + AOC = 180. As we said before, this is α+α + angle(AOC) = 180, or 2*α + angle(AOC) = 180.
Likewise, in triangle OCB, angle(OCB) + angle(OBC) + angle(COB) = 180. As noted, we can say β+β + angle(COB) = 180, or 2*β + angle(COB) = 180.
First Proof of Thales’ Theorem
We also see that triangle ACB is a triangle, so its interior angles add up to 180 also. So angle(BAC) + angle(ABC) + angle(ACB) = 180. That was α+β + angle(ACB) = 180.
But clearly angle(ACB) is the sum of angle(ACO) + angle(OCB), which are values α and β. So α+β + α+β = 180. So 2*(α+β) = 180. Dividing both sides by 2 shows that α+β = 90. This is precisely what Thales’ Theorem says, so it is proven.
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