Just when you see <start> then scan 3 numbers into double. You have the line content in line variable, you can use strtod to scan string into double. You can even use sscanf(line, "<start>%lf %lf %lf</start>", &x, &y, &z);, but using strtod is better for error handling.
#define _GNU_SOURCE
#include <stdio.h>
#include <string.h>
#include <errno.h>
#include <stddef.h>
#include <stdlib.h>
#define MAX_LINE_LENGTH 1024
void  print_lines(FILE *stream) {
    double a, b, c;
    char line[MAX_LINE_LENGTH];
    while (fgets(line, MAX_LINE_LENGTH, stream) != NULL) {
        char *pnt;
        // locate substring `<start>` in the line
        if ((pnt = strstr(line, "<start>") != NULL)) {
            // advance pnt to point to the characters after the `<start>`
            pnt = &pnt[sizeof("<start>") - 1];
            char *pntend;
            // scan first number
            a = strtod(pnt, &pntend);
            if (pnt == pntend) {
                fprintf(stderr, "Error converting a value.\n");
                // well, handle error some better way than ignoring.
            }
            pnt = pntend;
            // scan second number
            b = strtod(pnt, &pntend);
            if (pnt == pntend) {
                fprintf(stderr, "Error converting a value.\n");
                // well, handle error some better way than ignoring.
            }
            pnt = pntend;
            // scan third number
            c = strtod(pnt, &pntend);
            if (pnt == pntend) {
                fprintf(stderr, "Error converting a value.\n");
                // well, handle error some better way than ignoring.
            }
            printf("Read values are %lf %lf %lf\n", a, b, c);
        } else {
            // normal line
            //fputs(line, stdout);
        }
    }
}
int main()
{
    print_lines(stdin);
    return 0;
}