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The centers A, B, C of three congruent circles that have no common points do not lie on the same straight line. From points A, B, C, the six tangents shown in the figure are placed on the circles that enclose a convex hexagon.

Prove: The sums of the lengths of three pairs of not immediately adjacent sides of this hexagon are equal, i.e. | PQ | + | RS | + | TU | = | QR | + | ST | + | UP |

Answer :

MrRoyalgo

Mathematical proofs are used to determine if a mathematical statement is true or not.

See below for the proof of: [tex]\mathbf{PQ + RS + TU = QR + ST + UP}[/tex]

From the question (see attachment), we understand that: the three circles are congruent

This means that:

  • The radii of the three circles are the same
  • The length of the tangents are the same

Using the above highlights, we can conclude that:

  • [tex]\mathbf{UP = TU}[/tex]
  • [tex]\mathbf{ST =PQ}[/tex]
  • [tex]\mathbf{QR = RS}[/tex]

Add ST to both sides of [tex]\mathbf{UP = TU}[/tex]

[tex]\mathbf{ST + UP = ST + TU}[/tex]

By substitution property of equality; substitute [tex]\mathbf{ST =PQ}[/tex]

[tex]\mathbf{ST + UP = PQ + TU}[/tex]

Add QR to both sides of [tex]\mathbf{ST + UP = PQ + TU}[/tex]

[tex]\mathbf{QR + ST + UP = QR + PQ + TU}[/tex]

By substitution property of equality; substitute [tex]\mathbf{QR = RS}[/tex]

[tex]\mathbf{QR + ST + UP = RS + PQ + TU}[/tex]

Rewrite as:

[tex]\mathbf{QR + ST + UP =PQ + RS + TU}[/tex]

Hence, the sums of the lengths of three pairs of not immediately adjacent sides of this hexagon have been proved to be equal

Read more about mathematical proofs at:

https://brainly.com/question/843621

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