Maths Puzzle

I came across this puzzle in a book.

If everyone in a room shakes hands with every other person in the room, and there are 66 handshakes, how many people are in the room.

I can work out the answer logically by trial and error, and in a backwards way, but I would like to know the mathematical way to do it.

So if there are 3 people, I can call them A, B and C, and write down that A and B shake hands, A and C shake hands, and B and C shake hands, so there are 3 handshakes.
And I can continue like this with any number of people,
and eventually will come up with the answer.

2 people: A,B
AB = 1

3 people: A,B,C
AB AC = 2
BC = 1
Total = 3

4 people: A,B,C,D
AB AC AD = 3
BC BD = 2
Total = 5

5 people: A,B,C,D,E
AB AC AD AE = 4
BC BD BE = 3
CD CE = 2
DE = 1
Total = 10

6 people: A,B,C,D,E,F
AB AC AD AE AF = 5
BC BD BE BF = 4
CD CE CF = 3
DE DF = 2
EF = 1
Total = 15

7 people: A,B,C,D,E,F,G
AB AC AD AE AF AG = 6
BC BD BE BF BG = 5
CD CE CF CG = 4
DE DF DG = 3
EF EG = 2
FG = 1
Total = 21


12 people: A,B,C,D,E,F,G,H,J,K,L,M
AB AC AD AE AF AG AH AJ AK AL AM = 11
BC BD BE BF BG BH BJ BK AL AM = 10
CD CE CF CG CH CJ CK CL CM = 9
DE DF DG DH DJ DK DL DM = 8
EF EG EH EJ EK EL EM = 7
FG FH FJ FK FL FM = 6
GH GJ GK GL GM = 5
HJ HK HL HM = 4
JK JL JM = 3
KL KM = 2
LM = 1
Total =66

But I would like to know how to solve this mathematically by starting with the total number of handshakes (66) and finding the number of people in the room (12).