## Sunday, February 26, 2023

### Geometry Problem 1514: Discover the Secret to Finding Distances in Regular Hexagons with Interior Squares. Difficulty Level: High School.

Geometry Problem 1514. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

Details: Click on the figure below.

More Details

To post a solution to this problem click Comment underneath the post, or click into the line that says, “Enter Comment.” Type what you want to say and press Publish to post your solution.

1. GK = 6V2 ( square root) It will be explained in the sketch.

1. In Tr. GKH, < GKH > 90 so GH > GK and so GK < 6 but you say GK = 6.sqrt.2 > 6. Hence the answer is incorrect

2. https://photos.app.goo.gl/4AzheNU5tkxHPCtt8

3. This and some of previous are for beginners (students) : https://photos.app.goo.gl/upPuhDvxkQ8sj54B8

2. Since < AFE = 120 and < MFE = 90, <AFM = 30.
So <GFM = < GFA - < AFM = 45 - 30 =15 and < GFK = < NFM - < GFM = 45 - 15 = 30.

Hence Tr. GFK is 30-60-90 and so
GK = GF/2 = (6.sqrt.2)/2 which is the same as 6 / sqrt.2

Sumith Peiris
Moratuwa
Sri Lanka

3. Some interesting results

1) Tr.s ABG, AFM, FHE, END, GHM, MHN are all congruent and isosceles 75-30-75
2) B,G,K are collinear
3) Tr.s FEN & BKF are right isosceles

4. Not for post, just take a look, if square ASTB is added, then P, M, S are interesting points. https://photos.app.goo.gl/8xKYnMSYxAXuR9Jw7