In the adjoining equilateral triangle PQR , X , Y and Z are the middle points of the sides PQ , QR and RP respectively. Prove that XYZ is also an equilateral triangle.~Thanks in advance !

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TheAnimeGirl

High School

In the adjoining equilateral triangle PQR , X , Y and Z are the middle points of the sides PQ , QR and RP respectively. Prove that XYZ is also an equilateral triangle.

~Thanks in advance !

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(Based on todays review)

xKelvin

Answer:
See Below
Step-by-step explanation:
Statements: Reasons:
$1)\text{ } \Delta PQR\text{ is equilateral}$ Given
$2)\text{ }PQ=QR=RP$ Definition of Equilateral
$3)\text{ } X, Y, Z\text{ are the midpoints of } PQ, QR, RP$ Given
$4)\text{ } PX=XQ$ Definition of Midpoint
$5)PQ=PX+XQ$ Segment Addition
$6)\text{ } PQ=2XQ$ Substitution
$7)\text{ } QY=YR$ Definition of Midpoint
$8)\text{ }QR=QY+YR$ Segment Addition
$9)\text{ } QR=2QY$ Substitution
$10)\text{ } 2XQ=2QY$ Substitution
$11)\text{ } XQ=QY$ Division Property of Equality
$12)\text{ } XQ=PX=QY=YR$ Transitive Property
$13)\text{ } RZ=ZP$ Definition of Midpoint
$14)\text{ } RP=RZ+ZP$ Segment Addition
$15)\text{ } RP=2RZ$ Substitution
$16)\text{ } 2XQ=2RZ$ Substitution
$17)\text{ } XQ=RZ$ Substitution
$18)\text{ } XQ=PX=QY=YR=RZ=ZP$ Transitive Property
$19)\text{ } \angle Q\cong \angle R\cong \angle P$ Definition of Equilateral
$20)\text{ } \Delta XQY\cong \Delta YRQ \cong \Delta ZPX$ SAS Congruence
$21)\text{ } XY\cong YZ\cong ZX$ CPCTC
$22)\text{ } \Delta XYZ\text{ is equilateral}$ Equilateral Triangle Theorem
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