# Thread: Circumcenter of a triangle

1. ## Circumcenter of a triangle

I give up. I've tried different ways and to no avail, can someone please check my work, tell me where I went wrong and point me to the right direction?

Coordinates X- (3,36)
Y- (66, 69)
Z- (97, 74)

P – midpoint of ZY,

P 97 + 66 , 74 + 69
2 2

P 163 , 143
2 2

P ( 81.5 , 71.5 )

Q- midpoint of XZ,

Q 3 + 97 , 36 + 74
2 2

Q 100 , 110
2 2

Q (50 , 55)

R – midpoint of XY,

R 3 + 66 , 36 + 69
2 2

R 69 , 105
2 2

R (34.5 , 52.5)

Equation of line YR-
Y=mx + b

Y= 0.344x + b (97, 74)

Slope of line YR-

0.344

52.5 – 74 = -21.5 =
34.5 – 97 = -62.5 =

74 = 0.344 (97) + b

74 = 33.368 + b

b = 74 – 33.368

b = 40.632

Equation of line YR-

Y= 0.344x + 40.632

Slope of line YQ-

0.875

55 – 69 = -14 =
50 – 66 = -16 =

Equation of line YQ-
Y= 0.875x + b (66, 69)

69 = 0.875 (66) + b

69 = 57.75 + b

b = 69 – 57.75

b = 11.25

Equation of line YQ-

Y= 0.875x + 11.25

Equation of line XP-
Y = 0.452x + b ( 3, 36 )

Slope of line XP-
0.452

71.5 – 36 = 35.5 =
81.5 – 3 = 78.5 =

36 = 0.452 (3) + b

36 = 1.356 + b

b = 36 -1.356

b = 34.644

Equation of line XP-

Y= 0.452x + 34.644

-y + (-0.875x) = -11.25
y – 0.452x = 34.644

0.423x = 23.394
0.423 = 0.423

X= 55.305

Y – 0.452 (55.305) = 34.644

Y – 24.998 = 34.644

Y = 59.642

Point B – (55.305 , 59.642)

The coordinates of the circumcenter in triangle XYZ are (55.305, 59.642)

2. I always mess up these posts... (I'm a bad helper) Sorry...

3. Originally Posted by integral
if you know the coordinates just plug it into the Pythagorean theorem and find the lengths of each side (I am assuming by circumstance you mean perimeter... I have never heard it called anything other)

$\displaystyle c= \sqrt{a^2+b^2}$
I don't know what you mean by perimeter but circumcenter is the intersection of the 3 perpendicular bisectors in a triangle.

4. It's alright integral, you tried.

I tried a new method;
-63/33 (34.5, 52.5)
y - 52.5 = -1.909(x - 34.5)
y - 52.5 = -1.909x + 65.861
y = -1.909x + 118.361
-.161 ( 81.5 , 71.5)
y - 71.5 = -.161(x - 71.5)
y - 71.5 = -.161x + 11.512
y = -.161x + 83.012

Is anything correct, at all?

I got weird coordinates, please someone give me a step by step?