A circle \Gamma cuts the sides of a equilateral triangle ABC at 6 distinct points. Specifically, \Gamma intersects AB at points D and E such that A, D, E, B lie in order. \Gamma intersects BC at points F and G such that B, F, G, C lie in order. \Gamma intersects CA at points H and I such that C, H, I, A lie in order. If |AD| =3, |DE| =39, |EB| = 6 and |FG| = 21 , what is the value of |HI|^2 ?
From the secant-secant rule,
FB*21 = 6*39, so FB=78/7
AD+DE+EB = 3+39+6 = 48
CG+21+FB = 48, so CG = 111/7
CH*HI = CG*CF = 111/7 * 258/7 = 28638/49
AI*HI = 3*39 = 117
AI+IH+HC = 48
117/HI + HI + 28638/49HI = 48
now "just" solve for HI
The fractions look nasty, so you better check my arithmetic.
Oops. I was multiplying the wrong segments:
FB(FB+21) = 6*45
FB = 9
FB+FG+CG=48 so CG = 18
CH*(CH+HI) = 18*39 = 702
AI*(AI+HI) = 3*42 = 126
AI+HI+CH = 48
Things look better now
Yes, solving gives HI^2= 792.
To find the value of |HI|^2, we can use the Power of a Point theorem. Here's how you can approach this problem:
Step 1: Draw the equilateral triangle ABC and label the given points D, E, F, G, H, and I. Remember that the intersection points D, E, F, G, H, and I lie on the circumference of circle Γ.
Step 2: Identify pairs of similar triangles that will help us find the lengths we need. In this case, we can see that triangle ADE is similar to triangle ABC, as well as triangle FGB and triangle ABC. This is because corresponding angles in an equilateral triangle are equal, and so the ratios of corresponding sides will also be equal.
Step 3: Using the given lengths, we can set up a proportion to find the length of |AB|. Since triangle ADE is similar to triangle ABC, we can write the proportion:
|AD| / |AB| = |DE| / |AC|
Substituting the given values:
3 / |AB| = 39 / |AC|
Simplifying the equation, we get:
|AB| = 3 / (39 / |AC|) = 3 * |AC| / 39 = |AC| / 13
Step 4: Now that we have the length of |AB| in terms of |AC|, we can use the similar triangles to find the length of |AC|. Based on triangle FGB being similar to triangle ABC, we can set up the proportion:
|FG| / |AB| = |GB| / |BC|
Substituting the given value:
21 / |AB| = |GB| / |BC|
We already found that |AB| = |AC| / 13. Substituting this into the equation:
21 / (|AC| / 13) = |GB| / |BC|
Simplifying the equation, we find:
|GB| / |BC| = 273 / |AC|
Step 5: Now, we can use the Power of a Point theorem. According to the theorem, the product of the lengths of the line segments that are cut by a circle from a secant and its exterior section are equal. In this case, we can use D and E as the secant points.
We have "|AD| * |DE| = |BD| * |BE|". Substituting the given lengths, we get:
3 * 39 = |BD| * 6
117 = 6 * |BD|
|BD| = 117 / 6 = 19.5
Step 6: We have now found |BD|, and using the similar triangles, we can find |BC|. Based on triangle FGB being similar to triangle ABC, we can set up the proportion:
|GB| / |AB| = |BC| / |AC|
We already found that |GB| / |BC| = 273 / |AC|. Substituting this into the equation, we get:
273 / |AC| = |BC| / |AC|
Simplifying the equation, we find:
|BC| = 273
Step 7: Now, we can use the similar triangles to find |HC| and |IC|. Based on triangle HIA being similar to triangle ABC, we can set up the proportions:
|AC| / |AI| = |BC| / |AB|
Substituting the previously found values of |BC| = 273 and |AB| = |AC| / 13, we get:
|AC| / |AI| = 273 / (|AC| / 13)
Simplifying the equation, we find:
|AI| = 13
Similarly, using triangle HIB being similar to triangle ABC, we can find that |BI| = 13.
Step 8: Now, we can calculate the length of |HI|. Since AH and AI are both radii of circle Γ, |AH| = |AI|, i.e., |AH| = |AI| = 13. Therefore, using the Pythagorean theorem, we find:
|HI|^2 = |AH|^2 + |AI|^2
Substituting the values |AH| = 13 and |AI| = 13, we get:
|HI|^2 = 13^2 + 13^2
Simplifying, we find:
|HI|^2 = 338
Therefore, the value of |HI|^2 is 338.